import Mathlib.Data.Set.Basic
import Mathlib.Data.Set.Lattice

import Bookshelf.Enderton.Set.Chapter_1
import Common.Logic.Basic
import Common.Set.Basic

/-! # Enderton.Chapter_2

Axioms and Operations
-/

namespace Enderton.Set.Chapter_2


/-- ### Exercise 3.1

Assume that `A` is the set of integers divisible by `4`. Similarly assume that
`B` and `C` are the sets of integers divisible by `9` and `10`, respectively.
What is in `A ∩ B ∩ C`?
-/
theorem 
exercise_3_1: ∀ {A B C : Set ℤ},
  A = {x | 4 ∣ x} → B = {x | 9 ∣ x} → C = {x | 10 ∣ x} → ∀ (x : ℤ), x ∈ A ∩ B ∩ C → 4 ∣ x ∧ 9 ∣ x ∧ 10 ∣ x
exercise_3_1 {
A: Set ℤ
A 
B: Set ℤ
B 
C: Set ℤ
C : 
Set: Type ?u.5 → Type ?u.5
Set 
ℤ: Type
ℤ}
  (
hA: A = {x | 4 ∣ x}
hA : 
A: Set ℤ
A = { 
x: ?m.16
x | 
Dvd.dvd: {α : Type ?u.18} → [self : Dvd α] → α → α → Prop
Dvd.dvd 
4: ?m.22
4 
x: ?m.16
x })
  (
hB: B = {x | 9 ∣ x}
hB : 
B: Set ℤ
B = { 
x: ?m.94
x | 
Dvd.dvd: {α : Type ?u.96} → [self : Dvd α] → α → α → Prop
Dvd.dvd 
9: ?m.100
9 
x: ?m.94
x })
  (
hC: C = {x | 10 ∣ x}
hC : 
C: Set ℤ
C = { 
x: ?m.163
x | 
Dvd.dvd: {α : Type ?u.165} → [self : Dvd α] → α → α → Prop
Dvd.dvd 
10: ?m.169
10 
x: ?m.163
x })
  : ∀ 
x: ?m.228
x ∈ (
A: Set ℤ
A ∩ 
B: Set ℤ
B ∩ 
C: Set ℤ
C), (
4: ?m.288
4 ∣ 
x: ?m.228
x) ∧ (
9: ?m.300
9 ∣ 
x: ?m.228
x) ∧ (
10: ?m.312
10 ∣ 
x: ?m.228
x) := by
Goals accomplished! 🐙
  
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

∀ (x : ℤ), x ∈ A ∩ B ∩ C → 4 ∣ x ∧ 9 ∣ x ∧ 10 ∣ xintro 
x: ℤ
x 
h: x ∈ A ∩ B ∩ C
h
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∩ C

4 ∣ x ∧ 9 ∣ x ∧ 10 ∣ x
  
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

∀ (x : ℤ), x ∈ A ∩ B ∩ C → 4 ∣ x ∧ 9 ∣ x ∧ 10 ∣ xrw [
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∩ C

4 ∣ x ∧ 9 ∣ x ∧ 10 ∣ x
Set.mem_inter_iff: ∀ {α : Type ?u.341} (x : α) (a b : Set α), x ∈ a ∩ b ↔ x ∈ a ∧ x ∈ b
Set.mem_inter_iff
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

4 ∣ x ∧ 9 ∣ x ∧ 10 ∣ x]
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

4 ∣ x ∧ 9 ∣ x ∧ 10 ∣ x at 
h: x ∈ A ∩ B ∩ C
h
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

4 ∣ x ∧ 9 ∣ x ∧ 10 ∣ x
  
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

∀ (x : ℤ), x ∈ A ∩ B ∩ C → 4 ∣ x ∧ 9 ∣ x ∧ 10 ∣ xhave ⟨⟨
ha: x ∈ A
ha, 
hb: x ∈ B
hb⟩, 
hc: x ∈ C
hc⟩ := 
h: x ∈ A ∩ B ∧ x ∈ C
h
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ C

4 ∣ x ∧ 9 ∣ x ∧ 10 ∣ x
  
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

∀ (x : ℤ), x ∈ A ∩ B ∩ C → 4 ∣ x ∧ 9 ∣ x ∧ 10 ∣ xrefine ⟨
?_: ?m.506
?_, ⟨
?_: ?m.515
?_, 
?_: ?m.516
?_⟩⟩
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ C

refine_1
4 ∣ x
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ C

refine_2
9 ∣ x
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ C

refine_3
10 ∣ x
  
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

∀ (x : ℤ), x ∈ A ∩ B ∩ C → 4 ∣ x ∧ 9 ∣ x ∧ 10 ∣ x·
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ C

refine_1
4 ∣ x 
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ C

refine_1
4 ∣ x
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ C

refine_2
9 ∣ x
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ C

refine_3
10 ∣ xrw [
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ C

refine_1
4 ∣ x
hA: A = {x | 4 ∣ x}
hA
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ {x | 4 ∣ x}

hb: x ∈ B

hc: x ∈ C

refine_1
4 ∣ x]
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ {x | 4 ∣ x}

hb: x ∈ B

hc: x ∈ C

refine_1
4 ∣ x at 
ha: x ∈ A
ha
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ {x | 4 ∣ x}

hb: x ∈ B

hc: x ∈ C

refine_1
4 ∣ x
    
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ C

refine_1
4 ∣ xexact 
Set.mem_setOf: ∀ {α : Type ?u.541} {a : α} {p : α → Prop}, a ∈ {x | p x} ↔ p a
Set.mem_setOf.
mp: ∀ {a b : Prop}, (a ↔ b) → a → b
mp 
ha: x ∈ {x | 4 ∣ x}
ha
Goals accomplished! 🐙
  
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

∀ (x : ℤ), x ∈ A ∩ B ∩ C → 4 ∣ x ∧ 9 ∣ x ∧ 10 ∣ x·
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ C

refine_2
9 ∣ x 
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ C

refine_2
9 ∣ x
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ C

refine_3
10 ∣ xrw [
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ C

refine_2
9 ∣ x
hB: B = {x | 9 ∣ x}
hB
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ {x | 9 ∣ x}

hc: x ∈ C

refine_2
9 ∣ x]
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ {x | 9 ∣ x}

hc: x ∈ C

refine_2
9 ∣ x at 
hb: x ∈ B
hb
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ {x | 9 ∣ x}

hc: x ∈ C

refine_2
9 ∣ x
    
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ C

refine_2
9 ∣ xexact 
Set.mem_setOf: ∀ {α : Type ?u.571} {a : α} {p : α → Prop}, a ∈ {x | p x} ↔ p a
Set.mem_setOf.
mp: ∀ {a b : Prop}, (a ↔ b) → a → b
mp 
hb: x ∈ {x | 9 ∣ x}
hb
Goals accomplished! 🐙
  
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

∀ (x : ℤ), x ∈ A ∩ B ∩ C → 4 ∣ x ∧ 9 ∣ x ∧ 10 ∣ x·
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ C

refine_3
10 ∣ x 
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ C

refine_3
10 ∣ xrw [
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ C

refine_3
10 ∣ x
hC: C = {x | 10 ∣ x}
hC
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ {x | 10 ∣ x}

refine_3
10 ∣ x]
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ {x | 10 ∣ x}

refine_3
10 ∣ x at 
hc: x ∈ C
hc
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ {x | 10 ∣ x}

refine_3
10 ∣ x
    
A, B, C: Set ℤ

hA: A = {x | 4 ∣ x}

hB: B = {x | 9 ∣ x}

hC: C = {x | 10 ∣ x}

x: ℤ

h: x ∈ A ∩ B ∧ x ∈ C

ha: x ∈ A

hb: x ∈ B

hc: x ∈ C

refine_3
10 ∣ xexact 
Set.mem_setOf: ∀ {α : Type ?u.595} {a : α} {p : α → Prop}, a ∈ {x | p x} ↔ p a
Set.mem_setOf.
mp: ∀ {a b : Prop}, (a ↔ b) → a → b
mp 
hc: x ∈ {x | 10 ∣ x}
hc
Goals accomplished! 🐙

/-- ### Exercise 3.2

Give an example of sets `A` and `B` for which `⋃ A = ⋃ B` but `A ≠ B`.
-/
theorem 
exercise_3_2: ∀ {A B : Set (Set ℕ)}, A = {{1}, {2}} → B = {{1, 2}} → ⋃₀ A = ⋃₀ B ∧ A ≠ B
exercise_3_2 {
A: Set (Set ℕ)
A 
B: Set (Set ℕ)
B : 
Set: Type ?u.626 → Type ?u.626
Set (
Set: Type ?u.623 → Type ?u.623
Set 
ℕ: Type
ℕ)}
  (
hA: A = {{1}, {2}}
hA : 
A: Set (Set ℕ)
A = {{
1: ?m.651
1}, {
2: ?m.693
2}}) (
hB: B = {{1, 2}}
hB : 
B: Set (Set ℕ)
B = {{
1: ?m.848
1, 
2: ?m.870
2}})
  : 
Set.sUnion: {α : Type ?u.977} → Set (Set α) → Set α
Set.sUnion 
A: Set (Set ℕ)
A = 
Set.sUnion: {α : Type ?u.983} → Set (Set α) → Set α
Set.sUnion 
B: Set (Set ℕ)
B ∧ 
A: Set (Set ℕ)
A ≠ 
B: Set (Set ℕ)
B := by
Goals accomplished! 🐙
  
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

⋃₀ A = ⋃₀ B ∧ A ≠ Bapply 
And.intro: ∀ {a b : Prop}, a → b → a ∧ b
And.intro
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

left
⋃₀ A = ⋃₀ B
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

right
A ≠ B
  
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

⋃₀ A = ⋃₀ B ∧ A ≠ B·
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

left
⋃₀ A = ⋃₀ B 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

left
⋃₀ A = ⋃₀ B
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

right
A ≠ Bshow ⋃₀ 
A: Set (Set ℕ)
A = ⋃₀ 
B: Set (Set ℕ)
B
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

left
⋃₀ A = ⋃₀ B
    
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

left
⋃₀ A = ⋃₀ Bext 
x: ?α
x
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

left.h
x ∈ ⋃₀ A ↔ x ∈ ⋃₀ B
    
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

left
⋃₀ A = ⋃₀ Bshow (∃ 
t: ?m.1057
t, 
t: ?m.1057
t ∈ 
A: Set (Set ℕ)
A ∧ 
x: ?α
x ∈ 
t: ?m.1057
t) ↔ ∃ 
t: ?m.1092
t, 
t: ?m.1092
t ∈ 
B: Set (Set ℕ)
B ∧ 
x: ?α
x ∈ 
t: ?m.1092
t
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

left.h
(∃ t, t ∈ A ∧ x ∈ t) ↔ ∃ t, t ∈ B ∧ x ∈ t
    
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

left
⋃₀ A = ⋃₀ Bapply 
Iff.intro: ∀ {a b : Prop}, (a → b) → (b → a) → (a ↔ b)
Iff.intro
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

left.h.mp
(∃ t, t ∈ A ∧ x ∈ t) → ∃ t, t ∈ B ∧ x ∈ t
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

left.h.mpr
(∃ t, t ∈ B ∧ x ∈ t) → ∃ t, t ∈ A ∧ x ∈ t
    
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

left
⋃₀ A = ⋃₀ B·
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

left.h.mp
(∃ t, t ∈ A ∧ x ∈ t) → ∃ t, t ∈ B ∧ x ∈ t 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

left.h.mp
(∃ t, t ∈ A ∧ x ∈ t) → ∃ t, t ∈ B ∧ x ∈ t
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

left.h.mpr
(∃ t, t ∈ B ∧ x ∈ t) → ∃ t, t ∈ A ∧ x ∈ tintro ⟨
t: Set ℕ
t, ⟨
ht: t ∈ A
ht, 
hx: x ∈ t
hx⟩⟩
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ A

hx: x ∈ t

left.h.mp
∃ t, t ∈ B ∧ x ∈ t
      
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

left.h.mp
(∃ t, t ∈ A ∧ x ∈ t) → ∃ t, t ∈ B ∧ x ∈ trw [
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ A

hx: x ∈ t

left.h.mp
∃ t, t ∈ B ∧ x ∈ t
hA: A = {{1}, {2}}
hA
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

left.h.mp
∃ t, t ∈ B ∧ x ∈ t]
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

left.h.mp
∃ t, t ∈ B ∧ x ∈ t at 
ht: t ∈ A
ht
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

left.h.mp
∃ t, t ∈ B ∧ x ∈ t
      
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

left.h.mp
(∃ t, t ∈ A ∧ x ∈ t) → ∃ t, t ∈ B ∧ x ∈ trefine ⟨{
1: ?m.1314
1, 
2: ?m.1336
2}, ⟨
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

left.h.mp
∃ t, t ∈ B ∧ x ∈ tby
Goals accomplished! 🐙 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

{1, 2} ∈ Brw [
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

{1, 2} ∈ B
hB: B = {{1, 2}}
hB
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

{1, 2} ∈ {{1, 2}}]
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

{1, 2} ∈ {{1, 2}};
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

{1, 2} ∈ {{1, 2}} 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

{1, 2} ∈ Bsimp
Goals accomplished! 🐙, 
?_: ?m.1372
?_⟩⟩
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

left.h.mp
x ∈ {1, 2}
      
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

left.h.mp
(∃ t, t ∈ A ∧ x ∈ t) → ∃ t, t ∈ B ∧ x ∈ tapply 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
ht: t ∈ {{1}, {2}}
ht
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

left.h.mp.left
t = {1} → x ∈ {1, 2}
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

left.h.mp.right
t ∈ {{2}} → x ∈ {1, 2} 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

left.h.mp
x ∈ {1, 2}<;>
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

left.h.mp.left
t = {1} → x ∈ {1, 2}
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

left.h.mp.right
t ∈ {{2}} → x ∈ {1, 2}
      
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

left.h.mp
x ∈ {1, 2}·
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

left.h.mp.right
t ∈ {{2}} → x ∈ {1, 2} 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

left.h.mp.left
t = {1} → x ∈ {1, 2}intro 
ht': t ∈ {{2}}
ht'
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

ht': t ∈ {{2}}

left.h.mp.right
x ∈ {1, 2}
        
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

left.h.mp.left
t = {1} → x ∈ {1, 2}rw [
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

ht': t = {1}

left.h.mp.left
x ∈ {1, 2}
ht': t ∈ {{2}}
ht'
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ {2}

ht': t ∈ {{2}}

left.h.mp.right
x ∈ {1, 2}]
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ {2}

ht': t ∈ {{2}}

left.h.mp.right
x ∈ {1, 2} at 
hx: x ∈ t
hx
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ {1}

ht': t = {1}

left.h.mp.left
x ∈ {1, 2}
        
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

left.h.mp.left
t = {1} → x ∈ {1, 2}rw [
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ {2}

ht': t ∈ {{2}}

left.h.mp.right
x ∈ {1, 2}
hx: x ∈ {2}
hx
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ {2}

ht': t ∈ {{2}}

left.h.mp.right
2 ∈ {1, 2}]
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ {1}

ht': t = {1}

left.h.mp.left
1 ∈ {1, 2}
        
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1}, {2}}

hx: x ∈ t

left.h.mp.left
t = {1} → x ∈ {1, 2}simp
Goals accomplished! 🐙
    
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

left
⋃₀ A = ⋃₀ B·
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

left.h.mpr
(∃ t, t ∈ B ∧ x ∈ t) → ∃ t, t ∈ A ∧ x ∈ t 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

left.h.mpr
(∃ t, t ∈ B ∧ x ∈ t) → ∃ t, t ∈ A ∧ x ∈ tintro ⟨
t: Set ℕ
t, ⟨
ht: t ∈ B
ht, 
hx: x ∈ t
hx⟩⟩
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ B

hx: x ∈ t

left.h.mpr
∃ t, t ∈ A ∧ x ∈ t
      
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

left.h.mpr
(∃ t, t ∈ B ∧ x ∈ t) → ∃ t, t ∈ A ∧ x ∈ trw [
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ B

hx: x ∈ t

left.h.mpr
∃ t, t ∈ A ∧ x ∈ t
hB: B = {{1, 2}}
hB
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ t

left.h.mpr
∃ t, t ∈ A ∧ x ∈ t]
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ t

left.h.mpr
∃ t, t ∈ A ∧ x ∈ t at 
ht: t ∈ B
ht
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ t

left.h.mpr
∃ t, t ∈ A ∧ x ∈ t
      
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

left.h.mpr
(∃ t, t ∈ B ∧ x ∈ t) → ∃ t, t ∈ A ∧ x ∈ trw [
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ t

left.h.mpr
∃ t, t ∈ A ∧ x ∈ t
ht: t ∈ {{1, 2}}
ht
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

left.h.mpr
∃ t, t ∈ A ∧ x ∈ t]
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

left.h.mpr
∃ t, t ∈ A ∧ x ∈ t at 
hx: x ∈ t
hx
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

left.h.mpr
∃ t, t ∈ A ∧ x ∈ t
      
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

left.h.mpr
(∃ t, t ∈ B ∧ x ∈ t) → ∃ t, t ∈ A ∧ x ∈ tapply 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
hx: x ∈ {1, 2}
hx
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

left.h.mpr.left
x = 1 → ∃ t, t ∈ A ∧ x ∈ t
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

left.h.mpr.right
x ∈ {2} → ∃ t, t ∈ A ∧ x ∈ t
      
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

left.h.mpr
(∃ t, t ∈ B ∧ x ∈ t) → ∃ t, t ∈ A ∧ x ∈ t·
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

left.h.mpr.left
x = 1 → ∃ t, t ∈ A ∧ x ∈ t 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

left.h.mpr.left
x = 1 → ∃ t, t ∈ A ∧ x ∈ t
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

left.h.mpr.right
x ∈ {2} → ∃ t, t ∈ A ∧ x ∈ tintro 
hx': x = 1
hx'
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x = 1

left.h.mpr.left
∃ t, t ∈ A ∧ x ∈ t
        
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

left.h.mpr.left
x = 1 → ∃ t, t ∈ A ∧ x ∈ texact ⟨{
1: ?m.2064
1}, ⟨
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x = 1

left.h.mpr.left
∃ t, t ∈ A ∧ x ∈ tby
Goals accomplished! 🐙 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x = 1

{1} ∈ Arw [
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x = 1

{1} ∈ A
hA: A = {{1}, {2}}
hA
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x = 1

{1} ∈ {{1}, {2}}]
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x = 1

{1} ∈ {{1}, {2}};
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x = 1

{1} ∈ {{1}, {2}} 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x = 1

{1} ∈ Asimp
Goals accomplished! 🐙, 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x = 1

left.h.mpr.left
∃ t, t ∈ A ∧ x ∈ tby
Goals accomplished! 🐙 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x = 1

x ∈ {1}rw [
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x = 1

x ∈ {1}
hx': x = 1
hx'
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x = 1

1 ∈ {1}]
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x = 1

1 ∈ {1};
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x = 1

1 ∈ {1} 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x = 1

x ∈ {1}simp
Goals accomplished! 🐙⟩⟩
Goals accomplished! 🐙
      
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

left.h.mpr
(∃ t, t ∈ B ∧ x ∈ t) → ∃ t, t ∈ A ∧ x ∈ t·
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

left.h.mpr.right
x ∈ {2} → ∃ t, t ∈ A ∧ x ∈ t 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

left.h.mpr.right
x ∈ {2} → ∃ t, t ∈ A ∧ x ∈ tintro 
hx': x ∈ {2}
hx'
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x ∈ {2}

left.h.mpr.right
∃ t, t ∈ A ∧ x ∈ t
        
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

left.h.mpr.right
x ∈ {2} → ∃ t, t ∈ A ∧ x ∈ texact ⟨{
2: ?m.2423
2}, ⟨
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x ∈ {2}

left.h.mpr.right
∃ t, t ∈ A ∧ x ∈ tby
Goals accomplished! 🐙 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x ∈ {2}

{2} ∈ Arw [
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x ∈ {2}

{2} ∈ A
hA: A = {{1}, {2}}
hA
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x ∈ {2}

{2} ∈ {{1}, {2}}]
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x ∈ {2}

{2} ∈ {{1}, {2}};
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x ∈ {2}

{2} ∈ {{1}, {2}} 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x ∈ {2}

{2} ∈ Asimp
Goals accomplished! 🐙, 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x ∈ {2}

left.h.mpr.right
∃ t, t ∈ A ∧ x ∈ tby
Goals accomplished! 🐙 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x ∈ {2}

x ∈ {2}rw [
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x ∈ {2}

x ∈ {2}
hx': x ∈ {2}
hx'
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x ∈ {2}

2 ∈ {2}]
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x ∈ {2}

2 ∈ {2};
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x ∈ {2}

2 ∈ {2} 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

x: ℕ

t: Set ℕ

ht: t ∈ {{1, 2}}

hx: x ∈ {1, 2}

hx': x ∈ {2}

x ∈ {2}simp
Goals accomplished! 🐙⟩⟩
Goals accomplished! 🐙
  
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

⋃₀ A = ⋃₀ B ∧ A ≠ B·
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

right
A ≠ B 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

right
A ≠ Bshow 
A: Set (Set ℕ)
A ≠ 
B: Set (Set ℕ)
B
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

right
A ≠ B
    -- Find an element that exists in `B` but not in `A`. Extensionality
    -- concludes the proof.
    
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

right
A ≠ Bintro 
h: A = B
h
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

h: A = B

right
False
    
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

right
A ≠ Brw [
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

h: A = B

right
False
hA: A = {{1}, {2}}
hA,
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

h: {{1}, {2}} = B

right
False 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

h: A = B

right
False
hB: B = {{1, 2}}
hB,
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

h: {{1}, {2}} = {{1, 2}}

right
False 
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

h: A = B

right
False
Set.ext_iff: ∀ {α : Type ?u.2788} {s t : Set α}, s = t ↔ ∀ (x : α), x ∈ s ↔ x ∈ t
Set.ext_iff
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

h: ∀ (x : Set ℕ), x ∈ {{1}, {2}} ↔ x ∈ {{1, 2}}

right
False]
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

h: ∀ (x : Set ℕ), x ∈ {{1}, {2}} ↔ x ∈ {{1, 2}}

right
False at 
h: {{1}, {2}} = B
h
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

h: ∀ (x : Set ℕ), x ∈ {{1}, {2}} ↔ x ∈ {{1, 2}}

right
False
    
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

right
A ≠ Bhave 
h₁: ?m.2812
h₁ := 
h: ∀ (x : Set ℕ), x ∈ {{1}, {2}} ↔ x ∈ {{1, 2}}
h {
1: ?m.2826
1, 
2: ?m.2848
2}
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

h: ∀ (x : Set ℕ), x ∈ {{1}, {2}} ↔ x ∈ {{1, 2}}

h₁: {1, 2} ∈ {{1}, {2}} ↔ {1, 2} ∈ {{1, 2}}

right
False
    
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

right
A ≠ Bsimp at 
h₁: ?m.2812
h₁
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

h: ∀ (x : Set ℕ), x ∈ {{1}, {2}} ↔ x ∈ {{1, 2}}

h₁: {1, 2} = {1}

right
False
    
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

right
A ≠ Brw [
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

h: ∀ (x : Set ℕ), x ∈ {{1}, {2}} ↔ x ∈ {{1, 2}}

h₁: {1, 2} = {1}

right
False
Set.ext_iff: ∀ {α : Type ?u.3306} {s t : Set α}, s = t ↔ ∀ (x : α), x ∈ s ↔ x ∈ t
Set.ext_iff
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

h: ∀ (x : Set ℕ), x ∈ {{1}, {2}} ↔ x ∈ {{1, 2}}

h₁: ∀ (x : ℕ), x ∈ {1, 2} ↔ x ∈ {1}

right
False]
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

h: ∀ (x : Set ℕ), x ∈ {{1}, {2}} ↔ x ∈ {{1, 2}}

h₁: ∀ (x : ℕ), x ∈ {1, 2} ↔ x ∈ {1}

right
False at 
h₁: {1, 2} = {1}
h₁
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

h: ∀ (x : Set ℕ), x ∈ {{1}, {2}} ↔ x ∈ {{1, 2}}

h₁: ∀ (x : ℕ), x ∈ {1, 2} ↔ x ∈ {1}

right
False
    
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

right
A ≠ Bhave 
h₂: ?m.3330
h₂ := 
h₁: ∀ (x : ℕ), x ∈ {1, 2} ↔ x ∈ {1}
h₁ 
2: ?m.3332
2
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

h: ∀ (x : Set ℕ), x ∈ {{1}, {2}} ↔ x ∈ {{1, 2}}

h₁: ∀ (x : ℕ), x ∈ {1, 2} ↔ x ∈ {1}

h₂: 2 ∈ {1, 2} ↔ 2 ∈ {1}

right
False
    
A, B: Set (Set ℕ)

hA: A = {{1}, {2}}

hB: B = {{1, 2}}

right
A ≠ Bsimp at 
h₂: ?m.3330
h₂
Goals accomplished! 🐙

/-- ### Exercise 3.3

Show that every member of a set `A` is a subset of `U A`. (This was stated as an
example in this section.)
-/
theorem 
exercise_3_3: ∀ {α : Type u_1} {A : Set (Set α)} (x : Set α), x ∈ A → x ⊆ ⋃₀ A
exercise_3_3 {
A: Set (Set α)
A : 
Set: Type ?u.3473 → Type ?u.3473
Set (
Set: Type ?u.3474 → Type ?u.3474
Set 
α: ?m.3470
α)}
  : ∀ 
x: ?m.3478
x ∈ 
A: Set (Set α)
A, 
x: ?m.3478
x ⊆ 
Set.sUnion: {α : Type ?u.3520} → Set (Set α) → Set α
Set.sUnion 
A: Set (Set α)
A := by
Goals accomplished! 🐙
  
α: Type u_1

A: Set (Set α)

∀ (x : Set α), x ∈ A → x ⊆ ⋃₀ Aintro 
x: Set α
x 
hx: x ∈ A
hx
α: Type u_1

A: Set (Set α)

x: Set α

hx: x ∈ A

x ⊆ ⋃₀ A
  
α: Type u_1

A: Set (Set α)

∀ (x : Set α), x ∈ A → x ⊆ ⋃₀ Ashow ∀ 
y: ?m.3547
y ∈ 
x: Set α
x, 
y: ?m.3547
y ∈ { 
a: ?m.3598
a | ∃ 
t: ?m.3603
t, 
t: ?m.3603
t ∈ 
A: Set (Set α)
A ∧ 
a: ?m.3598
a ∈ 
t: ?m.3603
t }
α: Type u_1

A: Set (Set α)

x: Set α

hx: x ∈ A

∀ (y : α), y ∈ x → y ∈ {a | ∃ t, t ∈ A ∧ a ∈ t}
  
α: Type u_1

A: Set (Set α)

∀ (x : Set α), x ∈ A → x ⊆ ⋃₀ Aintro 
y: α
y 
hy: y ∈ x
hy
α: Type u_1

A: Set (Set α)

x: Set α

hx: x ∈ A

y: α

hy: y ∈ x

y ∈ {a | ∃ t, t ∈ A ∧ a ∈ t}
  
α: Type u_1

A: Set (Set α)

∀ (x : Set α), x ∈ A → x ⊆ ⋃₀ Arw [
α: Type u_1

A: Set (Set α)

x: Set α

hx: x ∈ A

y: α

hy: y ∈ x

y ∈ {a | ∃ t, t ∈ A ∧ a ∈ t}
Set.mem_setOf_eq: ∀ {α : Type ?u.3680} {x : α} {p : α → Prop}, (x ∈ {y | p y}) = p x
Set.mem_setOf_eq
α: Type u_1

A: Set (Set α)

x: Set α

hx: x ∈ A

y: α

hy: y ∈ x

∃ t, t ∈ A ∧ y ∈ t]
α: Type u_1

A: Set (Set α)

x: Set α

hx: x ∈ A

y: α

hy: y ∈ x

∃ t, t ∈ A ∧ y ∈ t
  
α: Type u_1

A: Set (Set α)

∀ (x : Set α), x ∈ A → x ⊆ ⋃₀ Aexact ⟨
x: Set α
x, ⟨
hx: x ∈ A
hx, 
hy: y ∈ x
hy⟩⟩
Goals accomplished! 🐙

/-- ### Exercise 3.4

Show that if `A ⊆ B`, then `⋃ A ⊆ ⋃ B`.
-/
theorem 
exercise_3_4: ∀ {α : Type u_1} {A B : Set (Set α)}, A ⊆ B → ⋃₀ A ⊆ ⋃₀ B
exercise_3_4 (
h: A ⊆ B
h : 
A: ?m.3754
A ⊆ 
B: ?m.3766
B) : ⋃₀ 
A: ?m.3754
A ⊆ ⋃₀ 
B: ?m.3766
B := by
Goals accomplished! 🐙
  
α✝: Type u_1

A, B: Set (Set α✝)

h: A ⊆ B

⋃₀ A ⊆ ⋃₀ Bshow ∀ 
x: ?m.3832
x ∈ { 
a: ?m.3855
a | ∃ 
t: ?m.3860
t, 
t: ?m.3860
t ∈ 
A: Set (Set α✝)
A ∧ 
a: ?m.3855
a ∈ 
t: ?m.3860
t }, 
x: ?m.3832
x ∈ { 
a: ?m.3927
a | ∃ 
t: ?m.3932
t, 
t: ?m.3932
t ∈ 
B: Set (Set α✝)
B ∧ 
a: ?m.3927
a ∈ 
t: ?m.3932
t }
α✝: Type u_1

A, B: Set (Set α✝)

h: A ⊆ B

∀ (x : α✝), x ∈ {a | ∃ t, t ∈ A ∧ a ∈ t} → x ∈ {a | ∃ t, t ∈ B ∧ a ∈ t}
  
α✝: Type u_1

A, B: Set (Set α✝)

h: A ⊆ B

⋃₀ A ⊆ ⋃₀ Bintro 
x: α✝
x 
hx: x ∈ {a | ∃ t, t ∈ A ∧ a ∈ t}
hx
α✝: Type u_1

A, B: Set (Set α✝)

h: A ⊆ B

x: α✝

hx: x ∈ {a | ∃ t, t ∈ A ∧ a ∈ t}

x ∈ {a | ∃ t, t ∈ B ∧ a ∈ t}
  
α✝: Type u_1

A, B: Set (Set α✝)

h: A ⊆ B

⋃₀ A ⊆ ⋃₀ Brw [
α✝: Type u_1

A, B: Set (Set α✝)

h: A ⊆ B

x: α✝

hx: x ∈ {a | ∃ t, t ∈ A ∧ a ∈ t}

x ∈ {a | ∃ t, t ∈ B ∧ a ∈ t}
Set.mem_setOf_eq: ∀ {α : Type ?u.4012} {x : α} {p : α → Prop}, (x ∈ {y | p y}) = p x
Set.mem_setOf_eq
α✝: Type u_1

A, B: Set (Set α✝)

h: A ⊆ B

x: α✝

hx: ∃ t, t ∈ A ∧ x ∈ t

x ∈ {a | ∃ t, t ∈ B ∧ a ∈ t}]
α✝: Type u_1

A, B: Set (Set α✝)

h: A ⊆ B

x: α✝

hx: ∃ t, t ∈ A ∧ x ∈ t

x ∈ {a | ∃ t, t ∈ B ∧ a ∈ t} at 
hx: x ∈ {a | ∃ t, t ∈ A ∧ a ∈ t}
hx
α✝: Type u_1

A, B: Set (Set α✝)

h: A ⊆ B

x: α✝

hx: ∃ t, t ∈ A ∧ x ∈ t

x ∈ {a | ∃ t, t ∈ B ∧ a ∈ t}
  
α✝: Type u_1

A, B: Set (Set α✝)

h: A ⊆ B

⋃₀ A ⊆ ⋃₀ Bhave ⟨
t: Set α✝
t, ⟨
ht: t ∈ A
ht, 
hxt: x ∈ t
hxt⟩⟩ := 
hx: ∃ t, t ∈ A ∧ x ∈ t
hx
α✝: Type u_1

A, B: Set (Set α✝)

h: A ⊆ B

x: α✝

hx: ∃ t, t ∈ A ∧ x ∈ t

t: Set α✝

ht: t ∈ A

hxt: x ∈ t

x ∈ {a | ∃ t, t ∈ B ∧ a ∈ t}
  
α✝: Type u_1

A, B: Set (Set α✝)

h: A ⊆ B

⋃₀ A ⊆ ⋃₀ Brw [
α✝: Type u_1

A, B: Set (Set α✝)

h: A ⊆ B

x: α✝

hx: ∃ t, t ∈ A ∧ x ∈ t

t: Set α✝

ht: t ∈ A

hxt: x ∈ t

x ∈ {a | ∃ t, t ∈ B ∧ a ∈ t}
Set.mem_setOf_eq: ∀ {α : Type ?u.4187} {x : α} {p : α → Prop}, (x ∈ {y | p y}) = p x
Set.mem_setOf_eq
α✝: Type u_1

A, B: Set (Set α✝)

h: A ⊆ B

x: α✝

hx: ∃ t, t ∈ A ∧ x ∈ t

t: Set α✝

ht: t ∈ A

hxt: x ∈ t

∃ t, t ∈ B ∧ x ∈ t]
α✝: Type u_1

A, B: Set (Set α✝)

h: A ⊆ B

x: α✝

hx: ∃ t, t ∈ A ∧ x ∈ t

t: Set α✝

ht: t ∈ A

hxt: x ∈ t

∃ t, t ∈ B ∧ x ∈ t
  
α✝: Type u_1

A, B: Set (Set α✝)

h: A ⊆ B

⋃₀ A ⊆ ⋃₀ Bexact ⟨
t: Set α✝
t, ⟨
h: A ⊆ B
h 
ht: t ∈ A
ht, 
hxt: x ∈ t
hxt⟩⟩
Goals accomplished! 🐙

/-- ### Exercise 3.5

Assume that every member of `𝓐` is a subset of `B`. Show that `⋃ 𝓐 ⊆ B`.
-/
theorem 
exercise_3_5: ∀ {α : Type u_1} {𝓐 : Set (Set α)} {B : Set α}, (∀ (x : Set α), x ∈ 𝓐 → x ⊆ B) → ⋃₀ 𝓐 ⊆ B
exercise_3_5 (
h: ∀ (x : Set α✝), x ∈ 𝓐 → x ⊆ B
h : ∀ 
x: ?m.4343
x ∈ 
𝓐: ?m.4290
𝓐, 
x: ?m.4343
x ⊆ 
B: ?m.4339
B) : ⋃₀ 
𝓐: ?m.4290
𝓐 ⊆ 
B: ?m.4339
B := by
Goals accomplished! 🐙
  
α✝: Type u_1

𝓐: Set (Set α✝)

B: Set α✝

h: ∀ (x : Set α✝), x ∈ 𝓐 → x ⊆ B

⋃₀ 𝓐 ⊆ Bshow ∀ 
y: ?m.4452
y ∈ { 
a: ?m.4475
a | ∃ 
t: ?m.4480
t, 
t: ?m.4480
t ∈ 
𝓐: Set (Set α✝)
𝓐 ∧ 
a: ?m.4475
a ∈ 
t: ?m.4480
t }, 
y: ?m.4452
y ∈ 
B: Set α✝
B
α✝: Type u_1

𝓐: Set (Set α✝)

B: Set α✝

h: ∀ (x : Set α✝), x ∈ 𝓐 → x ⊆ B

∀ (y : α✝), y ∈ {a | ∃ t, t ∈ 𝓐 ∧ a ∈ t} → y ∈ B
  
α✝: Type u_1

𝓐: Set (Set α✝)

B: Set α✝

h: ∀ (x : Set α✝), x ∈ 𝓐 → x ⊆ B

⋃₀ 𝓐 ⊆ Bintro 
y: α✝
y 
hy: y ∈ {a | ∃ t, t ∈ 𝓐 ∧ a ∈ t}
hy
α✝: Type u_1

𝓐: Set (Set α✝)

B: Set α✝

h: ∀ (x : Set α✝), x ∈ 𝓐 → x ⊆ B

y: α✝

hy: y ∈ {a | ∃ t, t ∈ 𝓐 ∧ a ∈ t}

y ∈ B
  
α✝: Type u_1

𝓐: Set (Set α✝)

B: Set α✝

h: ∀ (x : Set α✝), x ∈ 𝓐 → x ⊆ B

⋃₀ 𝓐 ⊆ Brw [
α✝: Type u_1

𝓐: Set (Set α✝)

B: Set α✝

h: ∀ (x : Set α✝), x ∈ 𝓐 → x ⊆ B

y: α✝

hy: y ∈ {a | ∃ t, t ∈ 𝓐 ∧ a ∈ t}

y ∈ B
Set.mem_setOf_eq: ∀ {α : Type ?u.4585} {x : α} {p : α → Prop}, (x ∈ {y | p y}) = p x
Set.mem_setOf_eq
α✝: Type u_1

𝓐: Set (Set α✝)

B: Set α✝

h: ∀ (x : Set α✝), x ∈ 𝓐 → x ⊆ B

y: α✝

hy: ∃ t, t ∈ 𝓐 ∧ y ∈ t

y ∈ B]
α✝: Type u_1

𝓐: Set (Set α✝)

B: Set α✝

h: ∀ (x : Set α✝), x ∈ 𝓐 → x ⊆ B

y: α✝

hy: ∃ t, t ∈ 𝓐 ∧ y ∈ t

y ∈ B at 
hy: y ∈ {a | ∃ t, t ∈ 𝓐 ∧ a ∈ t}
hy
α✝: Type u_1

𝓐: Set (Set α✝)

B: Set α✝

h: ∀ (x : Set α✝), x ∈ 𝓐 → x ⊆ B

y: α✝

hy: ∃ t, t ∈ 𝓐 ∧ y ∈ t

y ∈ B
  
α✝: Type u_1

𝓐: Set (Set α✝)

B: Set α✝

h: ∀ (x : Set α✝), x ∈ 𝓐 → x ⊆ B

⋃₀ 𝓐 ⊆ Bhave ⟨
t: Set α✝
t, ⟨
ht𝓐: t ∈ 𝓐
ht𝓐, 
hyt: y ∈ t
hyt⟩⟩ := 
hy: ∃ t, t ∈ 𝓐 ∧ y ∈ t
hy
α✝: Type u_1

𝓐: Set (Set α✝)

B: Set α✝

h: ∀ (x : Set α✝), x ∈ 𝓐 → x ⊆ B

y: α✝

hy: ∃ t, t ∈ 𝓐 ∧ y ∈ t

t: Set α✝

ht𝓐: t ∈ 𝓐

hyt: y ∈ t

y ∈ B
  
α✝: Type u_1

𝓐: Set (Set α✝)

B: Set α✝

h: ∀ (x : Set α✝), x ∈ 𝓐 → x ⊆ B

⋃₀ 𝓐 ⊆ Bexact (
h: ∀ (x : Set α✝), x ∈ 𝓐 → x ⊆ B
h 
t: Set α✝
t 
ht𝓐: t ∈ 𝓐
ht𝓐) 
hyt: y ∈ t
hyt
Goals accomplished! 🐙

/-- ### Exercise 3.6a

Show that for any set `A`, `⋃ 𝓟 A = A`.
-/
theorem 
exercise_3_6a: ∀ {α : Type u_1} {A : Set α}, ⋃₀ (𝒫 A) = A
exercise_3_6a : ⋃₀ (𝒫 
A: ?m.4787
A) = 
A: ?m.4787
A := by
Goals accomplished! 🐙
  
α✝: Type u_1

A: Set α✝

⋃₀ (𝒫 A) = Ashow { 
a: ?m.4814
a | ∃ 
t: ?m.4819
t, 
t: ?m.4819
t ∈ { 
t: ?m.4830
t | 
t: ?m.4830
t ⊆ 
A: Set α✝
A } ∧ 
a: ?m.4814
a ∈ 
t: ?m.4819
t } = 
A: Set α✝
A
α✝: Type u_1

A: Set α✝

{a | ∃ t, t ∈ {t | t ⊆ A} ∧ a ∈ t} = A
  
α✝: Type u_1

A: Set α✝

⋃₀ (𝒫 A) = Aext 
x: ?α
x
α✝: Type u_1

A: Set α✝

x: α✝

h
x ∈ {a | ∃ t, t ∈ {t | t ⊆ A} ∧ a ∈ t} ↔ x ∈ A
  
α✝: Type u_1

A: Set α✝

⋃₀ (𝒫 A) = Aapply 
Iff.intro: ∀ {a b : Prop}, (a → b) → (b → a) → (a ↔ b)
Iff.intro
α✝: Type u_1

A: Set α✝

x: α✝

h.mp
x ∈ {a | ∃ t, t ∈ {t | t ⊆ A} ∧ a ∈ t} → x ∈ A
α✝: Type u_1

A: Set α✝

x: α✝

h.mpr
x ∈ A → x ∈ {a | ∃ t, t ∈ {t | t ⊆ A} ∧ a ∈ t}
  
α✝: Type u_1

A: Set α✝

⋃₀ (𝒫 A) = A·
α✝: Type u_1

A: Set α✝

x: α✝

h.mp
x ∈ {a | ∃ t, t ∈ {t | t ⊆ A} ∧ a ∈ t} → x ∈ A 
α✝: Type u_1

A: Set α✝

x: α✝

h.mp
x ∈ {a | ∃ t, t ∈ {t | t ⊆ A} ∧ a ∈ t} → x ∈ A
α✝: Type u_1

A: Set α✝

x: α✝

h.mpr
x ∈ A → x ∈ {a | ∃ t, t ∈ {t | t ⊆ A} ∧ a ∈ t}intro 
hx: ?a
hx
α✝: Type u_1

A: Set α✝

x: α✝

hx: x ∈ {a | ∃ t, t ∈ {t | t ⊆ A} ∧ a ∈ t}

h.mp
x ∈ A
    
α✝: Type u_1

A: Set α✝

x: α✝

h.mp
x ∈ {a | ∃ t, t ∈ {t | t ⊆ A} ∧ a ∈ t} → x ∈ Arw [
α✝: Type u_1

A: Set α✝

x: α✝

hx: x ∈ {a | ∃ t, t ∈ {t | t ⊆ A} ∧ a ∈ t}

h.mp
x ∈ A
Set.mem_setOf_eq: ∀ {α : Type ?u.4941} {x : α} {p : α → Prop}, (x ∈ {y | p y}) = p x
Set.mem_setOf_eq
α✝: Type u_1

A: Set α✝

x: α✝

hx: ∃ t, t ∈ {t | t ⊆ A} ∧ x ∈ t

h.mp
x ∈ A]
α✝: Type u_1

A: Set α✝

x: α✝

hx: ∃ t, t ∈ {t | t ⊆ A} ∧ x ∈ t

h.mp
x ∈ A at 
hx: ?a
hx
α✝: Type u_1

A: Set α✝

x: α✝

hx: ∃ t, t ∈ {t | t ⊆ A} ∧ x ∈ t

h.mp
x ∈ A
    
α✝: Type u_1

A: Set α✝

x: α✝

h.mp
x ∈ {a | ∃ t, t ∈ {t | t ⊆ A} ∧ a ∈ t} → x ∈ Ahave ⟨
t: Set α✝
t, ⟨
htl: t ∈ {t | t ⊆ A}
htl, 
htr: x ∈ t
htr⟩⟩ := 
hx: ∃ t, t ∈ {t | t ⊆ A} ∧ x ∈ t
hx
α✝: Type u_1

A: Set α✝

x: α✝

hx: ∃ t, t ∈ {t | t ⊆ A} ∧ x ∈ t

t: Set α✝

htl: t ∈ {t | t ⊆ A}

htr: x ∈ t

h.mp
x ∈ A
    
α✝: Type u_1

A: Set α✝

x: α✝

h.mp
x ∈ {a | ∃ t, t ∈ {t | t ⊆ A} ∧ a ∈ t} → x ∈ Arw [
α✝: Type u_1

A: Set α✝

x: α✝

hx: ∃ t, t ∈ {t | t ⊆ A} ∧ x ∈ t

t: Set α✝

htl: t ∈ {t | t ⊆ A}

htr: x ∈ t

h.mp
x ∈ A
Set.mem_setOf_eq: ∀ {α : Type ?u.5116} {x : α} {p : α → Prop}, (x ∈ {y | p y}) = p x
Set.mem_setOf_eq
α✝: Type u_1

A: Set α✝

x: α✝

hx: ∃ t, t ∈ {t | t ⊆ A} ∧ x ∈ t

t: Set α✝

htl: t ⊆ A

htr: x ∈ t

h.mp
x ∈ A]
α✝: Type u_1

A: Set α✝

x: α✝

hx: ∃ t, t ∈ {t | t ⊆ A} ∧ x ∈ t

t: Set α✝

htl: t ⊆ A

htr: x ∈ t

h.mp
x ∈ A at 
htl: t ∈ {t | t ⊆ A}
htl
α✝: Type u_1

A: Set α✝

x: α✝

hx: ∃ t, t ∈ {t | t ⊆ A} ∧ x ∈ t

t: Set α✝

htl: t ⊆ A

htr: x ∈ t

h.mp
x ∈ A
    
α✝: Type u_1

A: Set α✝

x: α✝

h.mp
x ∈ {a | ∃ t, t ∈ {t | t ⊆ A} ∧ a ∈ t} → x ∈ Aexact 
htl: t ⊆ A
htl 
htr: x ∈ t
htr
Goals accomplished! 🐙
  
α✝: Type u_1

A: Set α✝

⋃₀ (𝒫 A) = A·
α✝: Type u_1

A: Set α✝

x: α✝

h.mpr
x ∈ A → x ∈ {a | ∃ t, t ∈ {t | t ⊆ A} ∧ a ∈ t} 
α✝: Type u_1

A: Set α✝

x: α✝

h.mpr
x ∈ A → x ∈ {a | ∃ t, t ∈ {t | t ⊆ A} ∧ a ∈ t}intro 
hx: ?b
hx
α✝: Type u_1

A: Set α✝

x: α✝

hx: x ∈ A

h.mpr
x ∈ {a | ∃ t, t ∈ {t | t ⊆ A} ∧ a ∈ t}
    
α✝: Type u_1

A: Set α✝

x: α✝

h.mpr
x ∈ A → x ∈ {a | ∃ t, t ∈ {t | t ⊆ A} ∧ a ∈ t}rw [
α✝: Type u_1

A: Set α✝

x: α✝

hx: x ∈ A

h.mpr
x ∈ {a | ∃ t, t ∈ {t | t ⊆ A} ∧ a ∈ t}
Set.mem_setOf_eq: ∀ {α : Type ?u.5164} {x : α} {p : α → Prop}, (x ∈ {y | p y}) = p x
Set.mem_setOf_eq
α✝: Type u_1

A: Set α✝

x: α✝

hx: x ∈ A

h.mpr
∃ t, t ∈ {t | t ⊆ A} ∧ x ∈ t]
α✝: Type u_1

A: Set α✝

x: α✝

hx: x ∈ A

h.mpr
∃ t, t ∈ {t | t ⊆ A} ∧ x ∈ t
    
α✝: Type u_1

A: Set α✝

x: α✝

h.mpr
x ∈ A → x ∈ {a | ∃ t, t ∈ {t | t ⊆ A} ∧ a ∈ t}exact ⟨
A: Set α✝
A, ⟨
α✝: Type u_1

A: Set α✝

x: α✝

hx: x ∈ A

h.mpr
∃ t, t ∈ {t | t ⊆ A} ∧ x ∈ tby
Goals accomplished! 🐙 
α✝: Type u_1

A: Set α✝

x: α✝

hx: x ∈ A

A ∈ {t | t ⊆ A}rw [
α✝: Type u_1

A: Set α✝

x: α✝

hx: x ∈ A

A ∈ {t | t ⊆ A}
Set.mem_setOf_eq: ∀ {α : Type ?u.5204} {x : α} {p : α → Prop}, (x ∈ {y | p y}) = p x
Set.mem_setOf_eq
α✝: Type u_1

A: Set α✝

x: α✝

hx: x ∈ A

A ⊆ A]
Goals accomplished! 🐙, 
hx: ?b
hx⟩⟩
Goals accomplished! 🐙

/-- ### Exercise 3.6b

Show that `A ⊆ 𝓟 ⋃ A`. Under what conditions does equality hold?
-/
theorem 
exercise_3_6b: ∀ {α : Type u_1} {A : Set (Set α)}, A ⊆ 𝒫⋃₀ A ∧ (A = 𝒫⋃₀ A ↔ ∃ B, A = 𝒫 B)
exercise_3_6b
  : 
A: ?m.5291
A ⊆ 𝒫 (⋃₀ 
A: ?m.5291
A)
  ∧ (
A: ?m.5291
A = 𝒫 (⋃₀ 
A: ?m.5291
A) ↔ ∃ 
B: ?m.5332
B, 
A: ?m.5291
A = 𝒫 
B: ?m.5332
B) := by
Goals accomplished! 🐙
  
α✝: Type u_1

A: Set (Set α✝)

A ⊆ 𝒫⋃₀ A ∧ (A = 𝒫⋃₀ A ↔ ∃ B, A = 𝒫 B)apply 
And.intro: ∀ {a b : Prop}, a → b → a ∧ b
And.intro
α✝: Type u_1

A: Set (Set α✝)

left
A ⊆ 𝒫⋃₀ A
α✝: Type u_1

A: Set (Set α✝)

right
A = 𝒫⋃₀ A ↔ ∃ B, A = 𝒫 B
  
α✝: Type u_1

A: Set (Set α✝)

A ⊆ 𝒫⋃₀ A ∧ (A = 𝒫⋃₀ A ↔ ∃ B, A = 𝒫 B)·
α✝: Type u_1

A: Set (Set α✝)

left
A ⊆ 𝒫⋃₀ A 
α✝: Type u_1

A: Set (Set α✝)

left
A ⊆ 𝒫⋃₀ A
α✝: Type u_1

A: Set (Set α✝)

right
A = 𝒫⋃₀ A ↔ ∃ B, A = 𝒫 Bshow ∀ 
x: ?m.5390
x ∈ 
A: Set (Set α✝)
A, 
x: ?m.5390
x ∈ { 
t: ?m.5442
t | 
t: ?m.5442
t ⊆ ⋃₀ 
A: Set (Set α✝)
A }
α✝: Type u_1

A: Set (Set α✝)

left
∀ (x : Set α✝), x ∈ A → x ∈ {t | t ⊆ ⋃₀ A}
    
α✝: Type u_1

A: Set (Set α✝)

left
A ⊆ 𝒫⋃₀ Aintro 
x: Set α✝
x 
hx: x ∈ A
hx
α✝: Type u_1

A: Set (Set α✝)

x: Set α✝

hx: x ∈ A

left
x ∈ {t | t ⊆ ⋃₀ A}
    
α✝: Type u_1

A: Set (Set α✝)

left
A ⊆ 𝒫⋃₀ Arw [
α✝: Type u_1

A: Set (Set α✝)

x: Set α✝

hx: x ∈ A

left
x ∈ {t | t ⊆ ⋃₀ A}
Set.mem_setOf: ∀ {α : Type ?u.5506} {a : α} {p : α → Prop}, a ∈ {x | p x} ↔ p a
Set.mem_setOf
α✝: Type u_1

A: Set (Set α✝)

x: Set α✝

hx: x ∈ A

left
x ⊆ ⋃₀ A]
α✝: Type u_1

A: Set (Set α✝)

x: Set α✝

hx: x ∈ A

left
x ⊆ ⋃₀ A
    
α✝: Type u_1

A: Set (Set α✝)

left
A ⊆ 𝒫⋃₀ Aexact 
exercise_3_3: ∀ {α : Type ?u.5588} {A : Set (Set α)} (x : Set α), x ∈ A → x ⊆ ⋃₀ A
exercise_3_3 
x: Set α✝
x 
hx: x ∈ A
hx
Goals accomplished! 🐙
  
α✝: Type u_1

A: Set (Set α✝)

A ⊆ 𝒫⋃₀ A ∧ (A = 𝒫⋃₀ A ↔ ∃ B, A = 𝒫 B)·
α✝: Type u_1

A: Set (Set α✝)

right
A = 𝒫⋃₀ A ↔ ∃ B, A = 𝒫 B 
α✝: Type u_1

A: Set (Set α✝)

right
A = 𝒫⋃₀ A ↔ ∃ B, A = 𝒫 Bapply 
Iff.intro: ∀ {a b : Prop}, (a → b) → (b → a) → (a ↔ b)
Iff.intro
α✝: Type u_1

A: Set (Set α✝)

right.mp
A = 𝒫⋃₀ A → ∃ B, A = 𝒫 B
α✝: Type u_1

A: Set (Set α✝)

right.mpr
(∃ B, A = 𝒫 B) → A = 𝒫⋃₀ A
    
α✝: Type u_1

A: Set (Set α✝)

right
A = 𝒫⋃₀ A ↔ ∃ B, A = 𝒫 B·
α✝: Type u_1

A: Set (Set α✝)

right.mp
A = 𝒫⋃₀ A → ∃ B, A = 𝒫 B 
α✝: Type u_1

A: Set (Set α✝)

right.mp
A = 𝒫⋃₀ A → ∃ B, A = 𝒫 B
α✝: Type u_1

A: Set (Set α✝)

right.mpr
(∃ B, A = 𝒫 B) → A = 𝒫⋃₀ Aintro 
hA: ?a
hA
α✝: Type u_1

A: Set (Set α✝)

hA: A = 𝒫⋃₀ A

right.mp
∃ B, A = 𝒫 B
      
α✝: Type u_1

A: Set (Set α✝)

right.mp
A = 𝒫⋃₀ A → ∃ B, A = 𝒫 Bexact ⟨⋃₀ 
A: Set (Set α✝)
A, 
hA: ?a
hA⟩
Goals accomplished! 🐙
    
α✝: Type u_1

A: Set (Set α✝)

right
A = 𝒫⋃₀ A ↔ ∃ B, A = 𝒫 B·
α✝: Type u_1

A: Set (Set α✝)

right.mpr
(∃ B, A = 𝒫 B) → A = 𝒫⋃₀ A 
α✝: Type u_1

A: Set (Set α✝)

right.mpr
(∃ B, A = 𝒫 B) → A = 𝒫⋃₀ Aintro ⟨
B: Set α✝
B, 
hB: A = 𝒫 B
hB⟩
α✝: Type u_1

A: Set (Set α✝)

B: Set α✝

hB: A = 𝒫 B

right.mpr
A = 𝒫⋃₀ A
      
α✝: Type u_1

A: Set (Set α✝)

right.mpr
(∃ B, A = 𝒫 B) → A = 𝒫⋃₀ Aconv =>
α✝: Type u_1

A: Set (Set α✝)

B: Set α✝

hB: A = 𝒫 B

𝒫 B 
α✝: Type u_1

A: Set (Set α✝)

B: Set α✝

hB: A = 𝒫 B

right.mpr
A = 𝒫⋃₀ Arhs
α✝: Type u_1

A: Set (Set α✝)

B: Set α✝

hB: A = 𝒫 B

𝒫⋃₀ A;
α✝: Type u_1

A: Set (Set α✝)

B: Set α✝

hB: A = 𝒫 B

𝒫⋃₀ A 
α✝: Type u_1

A: Set (Set α✝)

B: Set α✝

hB: A = 𝒫 B

A = 𝒫⋃₀ Arw [
α✝: Type u_1

A: Set (Set α✝)

B: Set α✝

hB: A = 𝒫 B

𝒫⋃₀ A
hB: A = 𝒫 B
hB,
α✝: Type u_1

A: Set (Set α✝)

B: Set α✝

hB: A = 𝒫 B

𝒫⋃₀ (𝒫 B) 
α✝: Type u_1

A: Set (Set α✝)

B: Set α✝

hB: A = 𝒫 B

𝒫⋃₀ A
exercise_3_6a: ∀ {α : Type ?u.5760} {A : Set α}, ⋃₀ (𝒫 A) = A
exercise_3_6a
α✝: Type u_1

A: Set (Set α✝)

B: Set α✝

hB: A = 𝒫 B

𝒫 B]
α✝: Type u_1

A: Set (Set α✝)

B: Set α✝

hB: A = 𝒫 B

𝒫 B
      
α✝: Type u_1

A: Set (Set α✝)

right.mpr
(∃ B, A = 𝒫 B) → A = 𝒫⋃₀ Aexact 
hB: A = 𝒫 B
hB
Goals accomplished! 🐙

/-- ### Exercise 3.7a

Show that for any sets `A` and `B`, `𝓟 A ∩ 𝓟 B = 𝓟 (A ∩ B)`.
-/
theorem 
exercise_3_7A: ∀ {α : Type u_1} {A B : Set α}, 𝒫 A ∩ 𝒫 B = 𝒫(A ∩ B)
exercise_3_7A
  : 𝒫 
A: ?m.5838
A ∩ 𝒫 
B: ?m.5856
B = 𝒫 (
A: ?m.5838
A ∩ 
B: ?m.5856
B) := by
Goals accomplished! 🐙
  
α✝: Type u_1

A, B: Set α✝

𝒫 A ∩ 𝒫 B = 𝒫(A ∩ B)suffices 𝒫 
A: Set α✝
A ∩ 𝒫 
B: Set α✝
B ⊆ 𝒫 (
A: Set α✝
A ∩ 
B: Set α✝
B) ∧ 𝒫 (
A: Set α✝
A ∩ 
B: Set α✝
B) ⊆ 𝒫 
A: Set α✝
A ∩ 𝒫 
B: Set α✝
B from
    
subset_antisymm: ∀ {α : Type ?u.6002} [inst : HasSubset α] {a b : α} [inst_1 : IsAntisymm α fun x x_1 => x ⊆ x_1], a ⊆ b → b ⊆ a → a = b
subset_antisymm 
this: 𝒫 A ∩ 𝒫 B ⊆ 𝒫(A ∩ B) ∧ 𝒫(A ∩ B) ⊆ 𝒫 A ∩ 𝒫 B
this.
left: ∀ {a b : Prop}, a ∧ b → a
left 
this: 𝒫 A ∩ 𝒫 B ⊆ 𝒫(A ∩ B) ∧ 𝒫(A ∩ B) ⊆ 𝒫 A ∩ 𝒫 B
this.
right: ∀ {a b : Prop}, a ∧ b → b
right
α✝: Type u_1

A, B: Set α✝

𝒫 A ∩ 𝒫 B ⊆ 𝒫(A ∩ B) ∧ 𝒫(A ∩ B) ⊆ 𝒫 A ∩ 𝒫 B
  
α✝: Type u_1

A, B: Set α✝

𝒫 A ∩ 𝒫 B = 𝒫(A ∩ B)apply 
And.intro: ∀ {a b : Prop}, a → b → a ∧ b
And.intro
α✝: Type u_1

A, B: Set α✝

left
𝒫 A ∩ 𝒫 B ⊆ 𝒫(A ∩ B)
α✝: Type u_1

A, B: Set α✝

right
𝒫(A ∩ B) ⊆ 𝒫 A ∩ 𝒫 B
  
α✝: Type u_1

A, B: Set α✝

𝒫 A ∩ 𝒫 B = 𝒫(A ∩ B)·
α✝: Type u_1

A, B: Set α✝

left
𝒫 A ∩ 𝒫 B ⊆ 𝒫(A ∩ B) 
α✝: Type u_1

A, B: Set α✝

left
𝒫 A ∩ 𝒫 B ⊆ 𝒫(A ∩ B)
α✝: Type u_1

A, B: Set α✝

right
𝒫(A ∩ B) ⊆ 𝒫 A ∩ 𝒫 Bunfold 
Set.powerset: {α : Type u_1} → Set α → Set (Set α)
Set.powerset
α✝: Type u_1

A, B: Set α✝

left
{t | t ⊆ A} ∩ {t | t ⊆ B} ⊆ {t | t ⊆ A ∩ B}
    
α✝: Type u_1

A, B: Set α✝

left
𝒫 A ∩ 𝒫 B ⊆ 𝒫(A ∩ B)intro 
x: Set α✝
x 
hx: x ∈ {t | t ⊆ A} ∩ {t | t ⊆ B}
hx
α✝: Type u_1

A, B, x: Set α✝

hx: x ∈ {t | t ⊆ A} ∩ {t | t ⊆ B}

left
x ∈ {t | t ⊆ A ∩ B}
    
α✝: Type u_1

A, B: Set α✝

left
𝒫 A ∩ 𝒫 B ⊆ 𝒫(A ∩ B)simp only [
Set.mem_inter_iff: ∀ {α : Type ?u.6079} (x : α) (a b : Set α), x ∈ a ∩ b ↔ x ∈ a ∧ x ∈ b
Set.mem_inter_iff, 
Set.mem_setOf_eq: ∀ {α : Type ?u.6105} {x : α} {p : α → Prop}, (x ∈ {y | p y}) = p x
Set.mem_setOf_eq] at 
hx: x ∈ {t | t ⊆ A} ∩ {t | t ⊆ B}
hx
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∧ x ⊆ B

left
x ∈ {t | t ⊆ A ∩ B}
    
α✝: Type u_1

A, B: Set α✝

left
𝒫 A ∩ 𝒫 B ⊆ 𝒫(A ∩ B)rwa [
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∧ x ⊆ B

left
x ∈ {t | t ⊆ A ∩ B}
Set.mem_setOf: ∀ {α : Type ?u.6229} {a : α} {p : α → Prop}, a ∈ {x | p x} ↔ p a
Set.mem_setOf,
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∧ x ⊆ B

left
x ⊆ A ∩ B 
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∧ x ⊆ B

left
x ∈ {t | t ⊆ A ∩ B}
Set.subset_inter_iff: ∀ {α : Type ?u.6240} {s t r : Set α}, r ⊆ s ∩ t ↔ r ⊆ s ∧ r ⊆ t
Set.subset_inter_iff
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∧ x ⊆ B

left
x ⊆ A ∧ x ⊆ B]
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∧ x ⊆ B

left
x ⊆ A ∧ x ⊆ B
  
α✝: Type u_1

A, B: Set α✝

𝒫 A ∩ 𝒫 B = 𝒫(A ∩ B)·
α✝: Type u_1

A, B: Set α✝

right
𝒫(A ∩ B) ⊆ 𝒫 A ∩ 𝒫 B 
α✝: Type u_1

A, B: Set α✝

right
𝒫(A ∩ B) ⊆ 𝒫 A ∩ 𝒫 Bunfold 
Set.powerset: {α : Type u_1} → Set α → Set (Set α)
Set.powerset
α✝: Type u_1

A, B: Set α✝

right
{t | t ⊆ A ∩ B} ⊆ {t | t ⊆ A} ∩ {t | t ⊆ B}
    
α✝: Type u_1

A, B: Set α✝

right
𝒫(A ∩ B) ⊆ 𝒫 A ∩ 𝒫 Bsimp
α✝: Type u_1

A, B: Set α✝

right
∀ (a : Set α✝), a ⊆ A → a ⊆ B → a ⊆ A
    
α✝: Type u_1

A, B: Set α✝

right
𝒫(A ∩ B) ⊆ 𝒫 A ∩ 𝒫 Bintro 
x: Set α✝
x 
hA: x ⊆ A
hA 
_: x ⊆ B
_
α✝: Type u_1

A, B, x: Set α✝

hA: x ⊆ A

a✝: x ⊆ B

right
x ⊆ A
    
α✝: Type u_1

A, B: Set α✝

right
𝒫(A ∩ B) ⊆ 𝒫 A ∩ 𝒫 Bexact 
hA: x ⊆ A
hA
Goals accomplished! 🐙

-- theorem false_of_false_iff_true : (false ↔ true) → false := by simp

/-- ### Exercise 3.7b (i)

Show that `𝓟 A ∪ 𝓟 B ⊆ 𝓟 (A ∪ B)`.
-/
theorem 
exercise_3_7b_i: ∀ {α : Type u_1} {A B : Set α}, 𝒫 A ∪ 𝒫 B ⊆ 𝒫(A ∪ B)
exercise_3_7b_i
  : 𝒫 
A: ?m.9908
A ∪ 𝒫 
B: ?m.9938
B ⊆ 𝒫 (
A: ?m.9908
A ∪ 
B: ?m.9938
B) := by
Goals accomplished! 🐙
  
α✝: Type u_1

A, B: Set α✝

𝒫 A ∪ 𝒫 B ⊆ 𝒫(A ∪ B)unfold 
Set.powerset: {α : Type u_1} → Set α → Set (Set α)
Set.powerset
α✝: Type u_1

A, B: Set α✝

{t | t ⊆ A} ∪ {t | t ⊆ B} ⊆ {t | t ⊆ A ∪ B}
  
α✝: Type u_1

A, B: Set α✝

𝒫 A ∪ 𝒫 B ⊆ 𝒫(A ∪ B)intro 
x: Set α✝
x 
hx: x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B}
hx
α✝: Type u_1

A, B, x: Set α✝

hx: x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B}

x ∈ {t | t ⊆ A ∪ B}
  
α✝: Type u_1

A, B: Set α✝

𝒫 A ∪ 𝒫 B ⊆ 𝒫(A ∪ B)simp at 
hx: x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B}
hx
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∨ x ⊆ B

x ∈ {t | t ⊆ A ∪ B}
  
α✝: Type u_1

A, B: Set α✝

𝒫 A ∪ 𝒫 B ⊆ 𝒫(A ∪ B)apply 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
hx: x ⊆ A ∨ x ⊆ B
hx
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∨ x ⊆ B

left
x ⊆ A → x ∈ {t | t ⊆ A ∪ B}
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∨ x ⊆ B

right
x ⊆ B → x ∈ {t | t ⊆ A ∪ B}
  
α✝: Type u_1

A, B: Set α✝

𝒫 A ∪ 𝒫 B ⊆ 𝒫(A ∪ B)·
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∨ x ⊆ B

left
x ⊆ A → x ∈ {t | t ⊆ A ∪ B} 
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∨ x ⊆ B

left
x ⊆ A → x ∈ {t | t ⊆ A ∪ B}
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∨ x ⊆ B

right
x ⊆ B → x ∈ {t | t ⊆ A ∪ B}intro 
hA: x ⊆ A
hA
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∨ x ⊆ B

hA: x ⊆ A

left
x ∈ {t | t ⊆ A ∪ B}
    
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∨ x ⊆ B

left
x ⊆ A → x ∈ {t | t ⊆ A ∪ B}rw [
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∨ x ⊆ B

hA: x ⊆ A

left
x ∈ {t | t ⊆ A ∪ B}
Set.mem_setOf_eq: ∀ {α : Type ?u.10631} {x : α} {p : α → Prop}, (x ∈ {y | p y}) = p x
Set.mem_setOf_eq
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∨ x ⊆ B

hA: x ⊆ A

left
x ⊆ A ∪ B]
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∨ x ⊆ B

hA: x ⊆ A

left
x ⊆ A ∪ B
    
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∨ x ⊆ B

left
x ⊆ A → x ∈ {t | t ⊆ A ∪ B}exact 
Set.subset_union_of_subset_left: ∀ {α : Type ?u.10705} {s t : Set α}, s ⊆ t → ∀ (u : Set α), s ⊆ t ∪ u
Set.subset_union_of_subset_left 
hA: x ⊆ A
hA 
B: Set α✝
B
Goals accomplished! 🐙
  
α✝: Type u_1

A, B: Set α✝

𝒫 A ∪ 𝒫 B ⊆ 𝒫(A ∪ B)·
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∨ x ⊆ B

right
x ⊆ B → x ∈ {t | t ⊆ A ∪ B} 
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∨ x ⊆ B

right
x ⊆ B → x ∈ {t | t ⊆ A ∪ B}intro 
hB: x ⊆ B
hB
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∨ x ⊆ B

hB: x ⊆ B

right
x ∈ {t | t ⊆ A ∪ B}
    
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∨ x ⊆ B

right
x ⊆ B → x ∈ {t | t ⊆ A ∪ B}rw [
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∨ x ⊆ B

hB: x ⊆ B

right
x ∈ {t | t ⊆ A ∪ B}
Set.mem_setOf_eq: ∀ {α : Type ?u.10719} {x : α} {p : α → Prop}, (x ∈ {y | p y}) = p x
Set.mem_setOf_eq
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∨ x ⊆ B

hB: x ⊆ B

right
x ⊆ A ∪ B]
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∨ x ⊆ B

hB: x ⊆ B

right
x ⊆ A ∪ B
    
α✝: Type u_1

A, B, x: Set α✝

hx: x ⊆ A ∨ x ⊆ B

right
x ⊆ B → x ∈ {t | t ⊆ A ∪ B}exact 
Set.subset_union_of_subset_right: ∀ {α : Type ?u.10790} {s u : Set α}, s ⊆ u → ∀ (t : Set α), s ⊆ t ∪ u
Set.subset_union_of_subset_right 
hB: x ⊆ B
hB 
A: Set α✝
A
Goals accomplished! 🐙

/-- ### Exercise 3.7b (ii)

Under what conditions does `𝓟 A ∪ 𝓟 B = 𝓟 (A ∪ B)`.?
-/
theorem 
exercise_3_7b_ii: ∀ {α : Type u_1} {A B : Set α}, 𝒫 A ∪ 𝒫 B = 𝒫(A ∪ B) ↔ A ⊆ B ∨ B ⊆ A
exercise_3_7b_ii
  : 𝒫 
A: ?m.10821
A ∪ 𝒫 
B: ?m.10839
B = 𝒫 (
A: ?m.10821
A ∪ 
B: ?m.10839
B) ↔ 
A: ?m.10821
A ⊆ 
B: ?m.10839
B ∨ 
B: ?m.10839
B ⊆ 
A: ?m.10821
A := by
Goals accomplished! 🐙
  
α✝: Type u_1

A, B: Set α✝

𝒫 A ∪ 𝒫 B = 𝒫(A ∪ B) ↔ A ⊆ B ∨ B ⊆ Aunfold 
Set.powerset: {α : Type u_1} → Set α → Set (Set α)
Set.powerset
α✝: Type u_1

A, B: Set α✝

{t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B} ↔ A ⊆ B ∨ B ⊆ A
  
α✝: Type u_1

A, B: Set α✝

𝒫 A ∪ 𝒫 B = 𝒫(A ∪ B) ↔ A ⊆ B ∨ B ⊆ Aapply 
Iff.intro: ∀ {a b : Prop}, (a → b) → (b → a) → (a ↔ b)
Iff.intro
α✝: Type u_1

A, B: Set α✝

mp
{t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B} → A ⊆ B ∨ B ⊆ A
α✝: Type u_1

A, B: Set α✝

mpr
A ⊆ B ∨ B ⊆ A → {t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B}
  
α✝: Type u_1

A, B: Set α✝

𝒫 A ∪ 𝒫 B = 𝒫(A ∪ B) ↔ A ⊆ B ∨ B ⊆ A·
α✝: Type u_1

A, B: Set α✝

mp
{t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B} → A ⊆ B ∨ B ⊆ A 
α✝: Type u_1

A, B: Set α✝

mp
{t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B} → A ⊆ B ∨ B ⊆ A
α✝: Type u_1

A, B: Set α✝

mpr
A ⊆ B ∨ B ⊆ A → {t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B}intro 
h: ?a
h
α✝: Type u_1

A, B: Set α✝

h: {t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B}

mp
A ⊆ B ∨ B ⊆ A
    
α✝: Type u_1

A, B: Set α✝

mp
{t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B} → A ⊆ B ∨ B ⊆ Aby_contra 
nh: ?m.11060
nh
α✝: Type u_1

A, B: Set α✝

h: {t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B}

nh: ¬(A ⊆ B ∨ B ⊆ A)

mp
False
    
α✝: Type u_1

A, B: Set α✝

mp
{t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B} → A ⊆ B ∨ B ⊆ Arw [
α✝: Type u_1

A, B: Set α✝

h: {t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B}

nh: ¬(A ⊆ B ∨ B ⊆ A)

mp
False
not_or_de_morgan: ∀ {p q : Prop}, ¬(p ∨ q) ↔ ¬p ∧ ¬q
not_or_de_morgan
α✝: Type u_1

A, B: Set α✝

h: {t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

mp
False]
α✝: Type u_1

A, B: Set α✝

h: {t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

mp
False at 
nh: ?m.11060
nh
α✝: Type u_1

A, B: Set α✝

h: {t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

mp
False
    
α✝: Type u_1

A, B: Set α✝

mp
{t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B} → A ⊆ B ∨ B ⊆ Ahave ⟨
a: α✝
a, 
hA: a ∈ A ∧ ¬a ∈ B
hA⟩ := 
Set.not_subset: ∀ {α : Type ?u.11087} {s t : Set α}, ¬s ⊆ t ↔ ∃ a, a ∈ s ∧ ¬a ∈ t
Set.not_subset.
mp: ∀ {a b : Prop}, (a ↔ b) → a → b
mp 
nh: ¬A ⊆ B ∧ ¬B ⊆ A
nh.
left: ∀ {a b : Prop}, a ∧ b → a
left
α✝: Type u_1

A, B: Set α✝

h: {t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

mp
False
    
α✝: Type u_1

A, B: Set α✝

mp
{t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B} → A ⊆ B ∨ B ⊆ Ahave ⟨
b: α✝
b, 
hB: b ∈ B ∧ ¬b ∈ A
hB⟩ := 
Set.not_subset: ∀ {α : Type ?u.11198} {s t : Set α}, ¬s ⊆ t ↔ ∃ a, a ∈ s ∧ ¬a ∈ t
Set.not_subset.
mp: ∀ {a b : Prop}, (a ↔ b) → a → b
mp 
nh: ¬A ⊆ B ∧ ¬B ⊆ A
nh.
right: ∀ {a b : Prop}, a ∧ b → b
right
α✝: Type u_1

A, B: Set α✝

h: {t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

mp
False
    
α✝: Type u_1

A, B: Set α✝

mp
{t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B} → A ⊆ B ∨ B ⊆ Arw [
α✝: Type u_1

A, B: Set α✝

h: {t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

mp
False
Set.ext_iff: ∀ {α : Type ?u.11312} {s t : Set α}, s = t ↔ ∀ (x : α), x ∈ s ↔ x ∈ t
Set.ext_iff
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

mp
False]
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

mp
False at 
h: ?a
h
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

mp
False
    
α✝: Type u_1

A, B: Set α✝

mp
{t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B} → A ⊆ B ∨ B ⊆ Ahave 
hz: ?m.11353
hz := 
h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}
h {
a: α✝
a, 
b: α✝
b}
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

hz: {a, b} ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ {a, b} ∈ {t | t ⊆ A ∪ B}

mp
False
    -- `hz` states that `{a, b} ⊆ A ∨ {a, b} ⊆ B ↔ {a, b} ⊆ A ∪ B`. We show the
    -- left-hand side is false but the right-hand side is true, yielding our
    -- contradiction.
    
α✝: Type u_1

A, B: Set α✝

mp
{t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B} → A ⊆ B ∨ B ⊆ Asuffices ¬({
a: α✝
a, 
b: α✝
b} ⊆ 
A: Set α✝
A ∨ {
a: α✝
a, 
b: α✝
b} ⊆ 
B: Set α✝
B) by
      
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

hz: {a, b} ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ {a, b} ∈ {t | t ⊆ A ∪ B}

mp
Falsehave 
hz₁: a ∈ A ∪ B
hz₁ : 
a: α✝
a ∈ 
A: Set α✝
A ∪ 
B: Set α✝
B := 
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

hz: {a, b} ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ {a, b} ∈ {t | t ⊆ A ∪ B}

this: ¬({a, b} ⊆ A ∨ {a, b} ⊆ B)

Falseby
Goals accomplished! 🐙
        
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

hz: {a, b} ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ {a, b} ∈ {t | t ⊆ A ∪ B}

this: ¬({a, b} ⊆ A ∨ {a, b} ⊆ B)

a ∈ A ∪ Brw [
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

hz: {a, b} ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ {a, b} ∈ {t | t ⊆ A ∪ B}

this: ¬({a, b} ⊆ A ∨ {a, b} ⊆ B)

a ∈ A ∪ B
Set.mem_union: ∀ {α : Type ?u.11607} (x : α) (a b : Set α), x ∈ a ∪ b ↔ x ∈ a ∨ x ∈ b
Set.mem_union
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

hz: {a, b} ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ {a, b} ∈ {t | t ⊆ A ∪ B}

this: ¬({a, b} ⊆ A ∨ {a, b} ⊆ B)

a ∈ A ∨ a ∈ B]
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

hz: {a, b} ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ {a, b} ∈ {t | t ⊆ A ∪ B}

this: ¬({a, b} ⊆ A ∨ {a, b} ⊆ B)

a ∈ A ∨ a ∈ B
        
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

hz: {a, b} ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ {a, b} ∈ {t | t ⊆ A ∪ B}

this: ¬({a, b} ⊆ A ∨ {a, b} ⊆ B)

a ∈ A ∪ Bexact 
Or.inl: ∀ {a b : Prop}, a → a ∨ b
Or.inl 
hA: a ∈ A ∧ ¬a ∈ B
hA.
left: ∀ {a b : Prop}, a ∧ b → a
left
Goals accomplished! 🐙
      
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

hz: {a, b} ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ {a, b} ∈ {t | t ⊆ A ∪ B}

this: ¬({a, b} ⊆ A ∨ {a, b} ⊆ B)

Falsehave 
hz₂: b ∈ A ∪ B
hz₂ : 
b: α✝
b ∈ 
A: Set α✝
A ∪ 
B: Set α✝
B := 
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

hz: {a, b} ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ {a, b} ∈ {t | t ⊆ A ∪ B}

this: ¬({a, b} ⊆ A ∨ {a, b} ⊆ B)

hz₁: a ∈ A ∪ B

Falseby
Goals accomplished! 🐙
        
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

hz: {a, b} ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ {a, b} ∈ {t | t ⊆ A ∪ B}

this: ¬({a, b} ⊆ A ∨ {a, b} ⊆ B)

hz₁: a ∈ A ∪ B

b ∈ A ∪ Brw [
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

hz: {a, b} ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ {a, b} ∈ {t | t ⊆ A ∪ B}

this: ¬({a, b} ⊆ A ∨ {a, b} ⊆ B)

hz₁: a ∈ A ∪ B

b ∈ A ∪ B
Set.mem_union: ∀ {α : Type ?u.11673} (x : α) (a b : Set α), x ∈ a ∪ b ↔ x ∈ a ∨ x ∈ b
Set.mem_union
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

hz: {a, b} ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ {a, b} ∈ {t | t ⊆ A ∪ B}

this: ¬({a, b} ⊆ A ∨ {a, b} ⊆ B)

hz₁: a ∈ A ∪ B

b ∈ A ∨ b ∈ B]
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

hz: {a, b} ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ {a, b} ∈ {t | t ⊆ A ∪ B}

this: ¬({a, b} ⊆ A ∨ {a, b} ⊆ B)

hz₁: a ∈ A ∪ B

b ∈ A ∨ b ∈ B
        
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

hz: {a, b} ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ {a, b} ∈ {t | t ⊆ A ∪ B}

this: ¬({a, b} ⊆ A ∨ {a, b} ⊆ B)

hz₁: a ∈ A ∪ B

b ∈ A ∪ Bexact 
Or.inr: ∀ {a b : Prop}, b → a ∨ b
Or.inr 
hB: b ∈ B ∧ ¬b ∈ A
hB.
left: ∀ {a b : Prop}, a ∧ b → a
left
Goals accomplished! 🐙
      
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

hz: {a, b} ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ {a, b} ∈ {t | t ⊆ A ∪ B}

this: ¬({a, b} ⊆ A ∨ {a, b} ⊆ B)

Falseexact 
absurd: ∀ {a : Prop} {b : Sort ?u.11693}, a → ¬a → b
absurd (
hz: {a, b} ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ {a, b} ∈ {t | t ⊆ A ∪ B}
hz.
mpr: ∀ {a b : Prop}, (a ↔ b) → b → a
mpr $ 
Set.mem_mem_imp_pair_subset: ∀ {α : Type ?u.11698} {A : Set α} {x y : α}, x ∈ A → y ∈ A → {x, y} ⊆ A
Set.mem_mem_imp_pair_subset 
hz₁: a ∈ A ∪ B
hz₁ 
hz₂: b ∈ A ∪ B
hz₂) 
this: ¬({a, b} ⊆ A ∨ {a, b} ⊆ B)
this
Goals accomplished! 🐙
    
α✝: Type u_1

A, B: Set α✝

mp
{t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B} → A ⊆ B ∨ B ⊆ Aintro 
hAB: {a, b} ⊆ A ∨ {a, b} ⊆ B
hAB
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

hz: {a, b} ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ {a, b} ∈ {t | t ⊆ A ∪ B}

hAB: {a, b} ⊆ A ∨ {a, b} ⊆ B

mp
False
    
α✝: Type u_1

A, B: Set α✝

mp
{t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B} → A ⊆ B ∨ B ⊆ Aexact 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
hAB: {a, b} ⊆ A ∨ {a, b} ⊆ B
hAB
      (fun 
y: ?m.11732
y => 
absurd: ∀ {a : Prop} {b : Sort ?u.11734}, a → ¬a → b
absurd (
y: ?m.11732
y $ show 
b: α✝
b ∈ {
a: α✝
a, 
b: α✝
b} by 
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

hz: {a, b} ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ {a, b} ∈ {t | t ⊆ A ∪ B}

hAB: {a, b} ⊆ A ∨ {a, b} ⊆ B

mp
Falsesimp
Goals accomplished! 🐙) 
hB: b ∈ B ∧ ¬b ∈ A
hB.
right: ∀ {a b : Prop}, a ∧ b → b
right)
      (fun 
y: ?m.12006
y => 
absurd: ∀ {a : Prop} {b : Sort ?u.12008}, a → ¬a → b
absurd (
y: ?m.12006
y $ show 
a: α✝
a ∈ {
a: α✝
a, 
b: α✝
b} by 
α✝: Type u_1

A, B: Set α✝

h: ∀ (x : Set α✝), x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}

nh: ¬A ⊆ B ∧ ¬B ⊆ A

a: α✝

hA: a ∈ A ∧ ¬a ∈ B

b: α✝

hB: b ∈ B ∧ ¬b ∈ A

hz: {a, b} ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ {a, b} ∈ {t | t ⊆ A ∪ B}

hAB: {a, b} ⊆ A ∨ {a, b} ⊆ B

mp
Falsesimp
Goals accomplished! 🐙) 
hA: a ∈ A ∧ ¬a ∈ B
hA.
right: ∀ {a b : Prop}, a ∧ b → b
right)
Goals accomplished! 🐙
  
α✝: Type u_1

A, B: Set α✝

𝒫 A ∪ 𝒫 B = 𝒫(A ∪ B) ↔ A ⊆ B ∨ B ⊆ A·
α✝: Type u_1

A, B: Set α✝

mpr
A ⊆ B ∨ B ⊆ A → {t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B} 
α✝: Type u_1

A, B: Set α✝

mpr
A ⊆ B ∨ B ⊆ A → {t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B}intro 
h: ?b
h
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

mpr
{t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B}
    
α✝: Type u_1

A, B: Set α✝

mpr
A ⊆ B ∨ B ⊆ A → {t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B}ext 
x: ?α
x
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

mpr.h
x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}
    
α✝: Type u_1

A, B: Set α✝

mpr
A ⊆ B ∨ B ⊆ A → {t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B}apply 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
h: ?b
h
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

mpr.h.left
A ⊆ B → (x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B})
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

mpr.h.right
B ⊆ A → (x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B})
    
α✝: Type u_1

A, B: Set α✝

mpr
A ⊆ B ∨ B ⊆ A → {t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B}·
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

mpr.h.left
A ⊆ B → (x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}) 
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

mpr.h.left
A ⊆ B → (x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B})
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

mpr.h.right
B ⊆ A → (x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B})intro 
hA: A ⊆ B
hA
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hA: A ⊆ B

mpr.h.left
x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}
      
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

mpr.h.left
A ⊆ B → (x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B})apply 
Iff.intro: ∀ {a b : Prop}, (a → b) → (b → a) → (a ↔ b)
Iff.intro
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hA: A ⊆ B

mpr.h.left.mp
x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} → x ∈ {t | t ⊆ A ∪ B}
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hA: A ⊆ B

mpr.h.left.mpr
x ∈ {t | t ⊆ A ∪ B} → x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B}
      
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

mpr.h.left
A ⊆ B → (x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B})·
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hA: A ⊆ B

mpr.h.left.mp
x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} → x ∈ {t | t ⊆ A ∪ B} 
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hA: A ⊆ B

mpr.h.left.mp
x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} → x ∈ {t | t ⊆ A ∪ B}
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hA: A ⊆ B

mpr.h.left.mpr
x ∈ {t | t ⊆ A ∪ B} → x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B}intro 
hx: ?a
hx
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hA: A ⊆ B

hx: x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B}

mpr.h.left.mp
x ∈ {t | t ⊆ A ∪ B}
        
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hA: A ⊆ B

mpr.h.left.mp
x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} → x ∈ {t | t ⊆ A ∪ B}exact 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
hx: ?a
hx
          (
Set.subset_union_of_subset_left: ∀ {α : Type ?u.12619} {s t : Set α}, s ⊆ t → ∀ (u : Set α), s ⊆ t ∪ u
Set.subset_union_of_subset_left · 
B: Set α✝
B)
          (
Set.subset_union_of_subset_right: ∀ {α : Type ?u.12634} {s u : Set α}, s ⊆ u → ∀ (t : Set α), s ⊆ t ∪ u
Set.subset_union_of_subset_right · 
A: Set α✝
A)
Goals accomplished! 🐙
      
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

mpr.h.left
A ⊆ B → (x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B})·
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hA: A ⊆ B

mpr.h.left.mpr
x ∈ {t | t ⊆ A ∪ B} → x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} 
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hA: A ⊆ B

mpr.h.left.mpr
x ∈ {t | t ⊆ A ∪ B} → x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B}intro 
hx: ?b
hx
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hA: A ⊆ B

hx: x ∈ {t | t ⊆ A ∪ B}

mpr.h.left.mpr
x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B}
        
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hA: A ⊆ B

mpr.h.left.mpr
x ∈ {t | t ⊆ A ∪ B} → x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B}refine 
Or.inr: ∀ {a b : Prop}, b → a ∨ b
Or.inr (
Set.Subset.trans: ∀ {α : Type ?u.12651} {a b c : Set α}, a ⊆ b → b ⊆ c → a ⊆ c
Set.Subset.trans 
hx: ?b
hx 
?_: ?m.12654 ⊆ ?m.12655
?_)
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hA: A ⊆ B

hx: x ∈ {t | t ⊆ A ∪ B}

mpr.h.left.mpr
A ∪ B ⊆ B
        
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hA: A ⊆ B

mpr.h.left.mpr
x ∈ {t | t ⊆ A ∪ B} → x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B}exact 
subset_of_eq: ∀ {α : Type ?u.12660} [inst : HasSubset α] {a b : α} [inst_1 : IsRefl α fun x x_1 => x ⊆ x_1], a = b → a ⊆ b
subset_of_eq (
Set.left_subset_union_eq_self: ∀ {α : Type ?u.12684} {A B : Set α}, A ⊆ B → A ∪ B = B
Set.left_subset_union_eq_self 
hA: A ⊆ B
hA)
Goals accomplished! 🐙
    
α✝: Type u_1

A, B: Set α✝

mpr
A ⊆ B ∨ B ⊆ A → {t | t ⊆ A} ∪ {t | t ⊆ B} = {t | t ⊆ A ∪ B}·
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

mpr.h.right
B ⊆ A → (x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}) 
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

mpr.h.right
B ⊆ A → (x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B})intro 
hB: B ⊆ A
hB
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hB: B ⊆ A

mpr.h.right
x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B}
      
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

mpr.h.right
B ⊆ A → (x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B})apply 
Iff.intro: ∀ {a b : Prop}, (a → b) → (b → a) → (a ↔ b)
Iff.intro
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hB: B ⊆ A

mpr.h.right.mp
x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} → x ∈ {t | t ⊆ A ∪ B}
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hB: B ⊆ A

mpr.h.right.mpr
x ∈ {t | t ⊆ A ∪ B} → x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B}
      
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

mpr.h.right
B ⊆ A → (x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B})·
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hB: B ⊆ A

mpr.h.right.mp
x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} → x ∈ {t | t ⊆ A ∪ B} 
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hB: B ⊆ A

mpr.h.right.mp
x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} → x ∈ {t | t ⊆ A ∪ B}
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hB: B ⊆ A

mpr.h.right.mpr
x ∈ {t | t ⊆ A ∪ B} → x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B}intro 
hx: ?a
hx
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hB: B ⊆ A

hx: x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B}

mpr.h.right.mp
x ∈ {t | t ⊆ A ∪ B}
        
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hB: B ⊆ A

mpr.h.right.mp
x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} → x ∈ {t | t ⊆ A ∪ B}exact 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
hx: ?a
hx
          (
Set.subset_union_of_subset_left: ∀ {α : Type ?u.12733} {s t : Set α}, s ⊆ t → ∀ (u : Set α), s ⊆ t ∪ u
Set.subset_union_of_subset_left · 
B: Set α✝
B)
          (
Set.subset_union_of_subset_right: ∀ {α : Type ?u.12748} {s u : Set α}, s ⊆ u → ∀ (t : Set α), s ⊆ t ∪ u
Set.subset_union_of_subset_right · 
A: Set α✝
A)
Goals accomplished! 🐙
      
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

mpr.h.right
B ⊆ A → (x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} ↔ x ∈ {t | t ⊆ A ∪ B})·
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hB: B ⊆ A

mpr.h.right.mpr
x ∈ {t | t ⊆ A ∪ B} → x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B} 
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hB: B ⊆ A

mpr.h.right.mpr
x ∈ {t | t ⊆ A ∪ B} → x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B}intro 
hx: ?b
hx
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hB: B ⊆ A

hx: x ∈ {t | t ⊆ A ∪ B}

mpr.h.right.mpr
x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B}
        
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hB: B ⊆ A

mpr.h.right.mpr
x ∈ {t | t ⊆ A ∪ B} → x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B}refine 
Or.inl: ∀ {a b : Prop}, a → a ∨ b
Or.inl (
Set.Subset.trans: ∀ {α : Type ?u.12765} {a b c : Set α}, a ⊆ b → b ⊆ c → a ⊆ c
Set.Subset.trans 
hx: ?b
hx 
?_: ?m.12768 ⊆ ?m.12769
?_)
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hB: B ⊆ A

hx: x ∈ {t | t ⊆ A ∪ B}

mpr.h.right.mpr
A ∪ B ⊆ A
        
α✝: Type u_1

A, B: Set α✝

h: A ⊆ B ∨ B ⊆ A

x: Set α✝

hB: B ⊆ A

mpr.h.right.mpr
x ∈ {t | t ⊆ A ∪ B} → x ∈ {t | t ⊆ A} ∪ {t | t ⊆ B}exact 
subset_of_eq: ∀ {α : Type ?u.12774} [inst : HasSubset α] {a b : α} [inst_1 : IsRefl α fun x x_1 => x ⊆ x_1], a = b → a ⊆ b
subset_of_eq (
Set.right_subset_union_eq_self: ∀ {α : Type ?u.12794} {A B : Set α}, B ⊆ A → A ∪ B = A
Set.right_subset_union_eq_self 
hB: B ⊆ A
hB)
Goals accomplished! 🐙

/-- ### Exercise 3.9

Give an example of sets `a` and `B` for which `a ∈ B` but `𝓟 a ∉ 𝓟 B`.
-/
theorem 
exercise_3_9: ∀ {a : Set ℕ} {B : Set (Set ℕ)}, a = {1} → B = {{1}} → a ∈ B ∧ ¬𝒫 a ∈ 𝒫 B
exercise_3_9 (
ha: a = {1}
ha : 
a: ?m.12845
a = {
1: unknown metavariable '?_uniq.12856'
1}) (
hB: B = {{1}}
hB : 
B: ?m.12895
B = {{
1: ?m.12963
1}})
  : 
a: ?m.12845
a ∈ 
B: ?m.12895
B ∧ 𝒫 
a: ?m.12845
a ∉ 𝒫 
B: ?m.12895
B := by
Goals accomplished! 🐙
  
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

a ∈ B ∧ ¬𝒫 a ∈ 𝒫 Bapply 
And.intro: ∀ {a b : Prop}, a → b → a ∧ b
And.intro
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

left
a ∈ B
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

right
¬𝒫 a ∈ 𝒫 B
  
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

a ∈ B ∧ ¬𝒫 a ∈ 𝒫 B·
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

left
a ∈ B 
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

left
a ∈ B
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

right
¬𝒫 a ∈ 𝒫 Brw [
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

left
a ∈ B
ha: a = {1}
ha,
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

left
{1} ∈ B 
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

left
a ∈ B
hB: B = {{1}}
hB
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

left
{1} ∈ {{1}}]
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

left
{1} ∈ {{1}}
    
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

left
a ∈ Bsimp
Goals accomplished! 🐙
  
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

a ∈ B ∧ ¬𝒫 a ∈ 𝒫 B·
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

right
¬𝒫 a ∈ 𝒫 B 
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

right
¬𝒫 a ∈ 𝒫 Bintro 
h: 𝒫 a ∈ 𝒫 B
h
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h: 𝒫 a ∈ 𝒫 B

right
False
    
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

right
¬𝒫 a ∈ 𝒫 Bhave 
h₁: 𝒫 a = {∅, {1}}
h₁ : 𝒫 
a: Set ℕ
a = {
∅: ?m.13378
∅, {
1: ?m.13472
1}} := 
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h: 𝒫 a ∈ 𝒫 B

right
Falseby
Goals accomplished! 🐙
      
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h: 𝒫 a ∈ 𝒫 B

𝒫 a = {∅, {1}}rw [
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h: 𝒫 a ∈ 𝒫 B

𝒫 a = {∅, {1}}
ha: a = {1}
ha
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h: 𝒫 a ∈ 𝒫 B

𝒫{1} = {∅, {1}}]
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h: 𝒫 a ∈ 𝒫 B

𝒫{1} = {∅, {1}}
      
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h: 𝒫 a ∈ 𝒫 B

𝒫 a = {∅, {1}}exact 
Set.powerset_singleton: ∀ {α : Type ?u.13657} (x : α), 𝒫{x} = {∅, {x}}
Set.powerset_singleton 
1: ?m.13660
1
Goals accomplished! 🐙
    
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

right
¬𝒫 a ∈ 𝒫 Bhave 
h₂: 𝒫 B = {∅, {{1}}}
h₂ : 𝒫 
B: Set (Set ℕ)
B = {
∅: ?m.13694
∅, {{
1: ?m.13800
1}}} := 
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h: 𝒫 a ∈ 𝒫 B

h₁: 𝒫 a = {∅, {1}}

right
Falseby
Goals accomplished! 🐙
      
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h: 𝒫 a ∈ 𝒫 B

h₁: 𝒫 a = {∅, {1}}

𝒫 B = {∅, {{1}}}rw [
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h: 𝒫 a ∈ 𝒫 B

h₁: 𝒫 a = {∅, {1}}

𝒫 B = {∅, {{1}}}
hB: B = {{1}}
hB
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h: 𝒫 a ∈ 𝒫 B

h₁: 𝒫 a = {∅, {1}}

𝒫{{1}} = {∅, {{1}}}]
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h: 𝒫 a ∈ 𝒫 B

h₁: 𝒫 a = {∅, {1}}

𝒫{{1}} = {∅, {{1}}}
      
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h: 𝒫 a ∈ 𝒫 B

h₁: 𝒫 a = {∅, {1}}

𝒫 B = {∅, {{1}}}exact 
Set.powerset_singleton: ∀ {α : Type ?u.14014} (x : α), 𝒫{x} = {∅, {x}}
Set.powerset_singleton {
1: ?m.14029
1}
Goals accomplished! 🐙
    
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

right
¬𝒫 a ∈ 𝒫 Brw [
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h: 𝒫 a ∈ 𝒫 B

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

right
False
h₁: 𝒫 a = {∅, {1}}
h₁,
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h: {∅, {1}} ∈ 𝒫 B

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

right
False 
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h: 𝒫 a ∈ 𝒫 B

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

right
False
h₂: 𝒫 B = {∅, {{1}}}
h₂
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h: {∅, {1}} ∈ {∅, {{1}}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

right
False]
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h: {∅, {1}} ∈ {∅, {{1}}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

right
False at 
h: 𝒫 a ∈ 𝒫 B
h
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h: {∅, {1}} ∈ {∅, {{1}}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

right
False
    
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

right
¬𝒫 a ∈ 𝒫 Bsimp at 
h: {∅, {1}} ∈ {∅, {{1}}}
h
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h: {∅, {1}} = ∅ ∨ ∅ = {1}

right
False
    
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

right
¬𝒫 a ∈ 𝒫 Bapply 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
h: {∅, {1}} = ∅ ∨ ∅ = {1}
h
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h: {∅, {1}} = ∅ ∨ ∅ = {1}

right.left
{∅, {1}} = ∅ → False
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h: {∅, {1}} = ∅ ∨ ∅ = {1}

right.right
∅ = {1} → False
    
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

right
¬𝒫 a ∈ 𝒫 B·
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h: {∅, {1}} = ∅ ∨ ∅ = {1}

right.left
{∅, {1}} = ∅ → False 
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h: {∅, {1}} = ∅ ∨ ∅ = {1}

right.left
{∅, {1}} = ∅ → False
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h: {∅, {1}} = ∅ ∨ ∅ = {1}

right.right
∅ = {1} → Falseintro 
h: {∅, {1}} = ∅
h
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h✝: {∅, {1}} = ∅ ∨ ∅ = {1}

h: {∅, {1}} = ∅

right.left
False
      
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h: {∅, {1}} = ∅ ∨ ∅ = {1}

right.left
{∅, {1}} = ∅ → Falserw [
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h✝: {∅, {1}} = ∅ ∨ ∅ = {1}

h: {∅, {1}} = ∅

right.left
False
Set.ext_iff: ∀ {α : Type ?u.14681} {s t : Set α}, s = t ↔ ∀ (x : α), x ∈ s ↔ x ∈ t
Set.ext_iff
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h✝: {∅, {1}} = ∅ ∨ ∅ = {1}

h: ∀ (x : Set ℕ), x ∈ {∅, {1}} ↔ x ∈ ∅

right.left
False]
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h✝: {∅, {1}} = ∅ ∨ ∅ = {1}

h: ∀ (x : Set ℕ), x ∈ {∅, {1}} ↔ x ∈ ∅

right.left
False at 
h: {∅, {1}} = ∅
h
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h✝: {∅, {1}} = ∅ ∨ ∅ = {1}

h: ∀ (x : Set ℕ), x ∈ {∅, {1}} ↔ x ∈ ∅

right.left
False
      
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h: {∅, {1}} = ∅ ∨ ∅ = {1}

right.left
{∅, {1}} = ∅ → Falsehave := 
h: ∀ (x : Set ℕ), x ∈ {∅, {1}} ↔ x ∈ ∅
h 
∅: ?m.14707
∅
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h✝: {∅, {1}} = ∅ ∨ ∅ = {1}

h: ∀ (x : Set ℕ), x ∈ {∅, {1}} ↔ x ∈ ∅

this: ∅ ∈ {∅, {1}} ↔ ∅ ∈ ∅

right.left
False
      
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h: {∅, {1}} = ∅ ∨ ∅ = {1}

right.left
{∅, {1}} = ∅ → Falsesimp at 
this: ?m.14705
this
Goals accomplished! 🐙
    
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

right
¬𝒫 a ∈ 𝒫 B·
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h: {∅, {1}} = ∅ ∨ ∅ = {1}

right.right
∅ = {1} → False 
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h: {∅, {1}} = ∅ ∨ ∅ = {1}

right.right
∅ = {1} → Falseintro 
h: ∅ = {1}
h
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h✝: {∅, {1}} = ∅ ∨ ∅ = {1}

h: ∅ = {1}

right.right
False
      
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h: {∅, {1}} = ∅ ∨ ∅ = {1}

right.right
∅ = {1} → Falserw [
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h✝: {∅, {1}} = ∅ ∨ ∅ = {1}

h: ∅ = {1}

right.right
False
Set.ext_iff: ∀ {α : Type ?u.14942} {s t : Set α}, s = t ↔ ∀ (x : α), x ∈ s ↔ x ∈ t
Set.ext_iff
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h✝: {∅, {1}} = ∅ ∨ ∅ = {1}

h: ∀ (x : ℕ), x ∈ ∅ ↔ x ∈ {1}

right.right
False]
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h✝: {∅, {1}} = ∅ ∨ ∅ = {1}

h: ∀ (x : ℕ), x ∈ ∅ ↔ x ∈ {1}

right.right
False at 
h: ∅ = {1}
h
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h✝: {∅, {1}} = ∅ ∨ ∅ = {1}

h: ∀ (x : ℕ), x ∈ ∅ ↔ x ∈ {1}

right.right
False
      
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h: {∅, {1}} = ∅ ∨ ∅ = {1}

right.right
∅ = {1} → Falsehave := 
h: ∀ (x : ℕ), x ∈ ∅ ↔ x ∈ {1}
h 
1: ?m.14968
1
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h✝: {∅, {1}} = ∅ ∨ ∅ = {1}

h: ∀ (x : ℕ), x ∈ ∅ ↔ x ∈ {1}

this: 1 ∈ ∅ ↔ 1 ∈ {1}

right.right
False
      
a: Set ℕ

B: Set (Set ℕ)

ha: a = {1}

hB: B = {{1}}

h₁: 𝒫 a = {∅, {1}}

h₂: 𝒫 B = {∅, {{1}}}

h: {∅, {1}} = ∅ ∨ ∅ = {1}

right.right
∅ = {1} → Falsesimp at 
this: ?m.14966
this
Goals accomplished! 🐙

/-- ### Exercise 3.10

Show that if `a ∈ B`, then `𝓟 a ∈ 𝓟 𝓟 ⋃ B`.
-/
theorem 
exercise_3_10: ∀ {α : Type u_1} {a : Set α} {B : Set (Set α)}, a ∈ B → 𝒫 a ∈ 𝒫 𝒫⋃₀ B
exercise_3_10 (
ha: a ∈ B
ha : 
a: ?m.15059
a ∈ 
B: ?m.15078
B)
  : 𝒫 
a: ?m.15059
a ∈ 𝒫 (𝒫 (⋃₀ 
B: ?m.15078
B)) := by
Goals accomplished! 🐙
  
α✝: Type u_1

a: Set α✝

B: Set (Set α✝)

ha: a ∈ B

𝒫 a ∈ 𝒫 𝒫⋃₀ Bhave 
h₁: ?m.15182
h₁ := 
exercise_3_3: ∀ {α : Type ?u.15183} {A : Set (Set α)} (x : Set α), x ∈ A → x ⊆ ⋃₀ A
exercise_3_3 
a: Set α✝
a 
ha: a ∈ B
ha
α✝: Type u_1

a: Set α✝

B: Set (Set α✝)

ha: a ∈ B

h₁: a ⊆ ⋃₀ B

𝒫 a ∈ 𝒫 𝒫⋃₀ B
  
α✝: Type u_1

a: Set α✝

B: Set (Set α✝)

ha: a ∈ B

𝒫 a ∈ 𝒫 𝒫⋃₀ Bhave 
h₂: ?m.15201
h₂ := 
Chapter_1.exercise_1_3: ∀ {α : Type ?u.15202} {B C : Set α}, B ⊆ C → 𝒫 B ⊆ 𝒫 C
Chapter_1.exercise_1_3 
h₁: ?m.15182
h₁
α✝: Type u_1

a: Set α✝

B: Set (Set α✝)

ha: a ∈ B

h₁: a ⊆ ⋃₀ B

h₂: 𝒫 a ⊆ 𝒫⋃₀ B

𝒫 a ∈ 𝒫 𝒫⋃₀ B
  
α✝: Type u_1

a: Set α✝

B: Set (Set α✝)

ha: a ∈ B

𝒫 a ∈ 𝒫 𝒫⋃₀ Bgeneralize hb : 𝒫 (⋃₀ 
B: Set (Set α✝)
B) = b
α✝: Type u_1

a: Set α✝

B: Set (Set α✝)

ha: a ∈ B

h₁: a ⊆ ⋃₀ B

h₂: 𝒫 a ⊆ 𝒫⋃₀ B

b: Set (Set α✝)

hb: 𝒫⋃₀ B = b

𝒫 a ∈ 𝒫 b
  
α✝: Type u_1

a: Set α✝

B: Set (Set α✝)

ha: a ∈ B

𝒫 a ∈ 𝒫 𝒫⋃₀ Bconv =>
α✝: Type u_1

a: Set α✝

B: Set (Set α✝)

ha: a ∈ B

h₁: a ⊆ ⋃₀ B

h₂: 𝒫 a ⊆ 𝒫⋃₀ B

b: Set (Set α✝)

hb: 𝒫⋃₀ B = b

{t | t ⊆ b} 
α✝: Type u_1

a: Set α✝

B: Set (Set α✝)

ha: a ∈ B

h₁: a ⊆ ⋃₀ B

h₂: 𝒫 a ⊆ 𝒫⋃₀ B

b: Set (Set α✝)

hb: 𝒫⋃₀ B = b

𝒫 a ∈ 𝒫 brhs
α✝: Type u_1

a: Set α✝

B: Set (Set α✝)

ha: a ∈ B

h₁: a ⊆ ⋃₀ B

h₂: 𝒫 a ⊆ 𝒫⋃₀ B

b: Set (Set α✝)

hb: 𝒫⋃₀ B = b

𝒫 b;
α✝: Type u_1

a: Set α✝

B: Set (Set α✝)

ha: a ∈ B

h₁: a ⊆ ⋃₀ B

h₂: 𝒫 a ⊆ 𝒫⋃₀ B

b: Set (Set α✝)

hb: 𝒫⋃₀ B = b

𝒫 b 
α✝: Type u_1

a: Set α✝

B: Set (Set α✝)

ha: a ∈ B

h₁: a ⊆ ⋃₀ B

h₂: 𝒫 a ⊆ 𝒫⋃₀ B

b: Set (Set α✝)

hb: 𝒫⋃₀ B = b

𝒫 a ∈ 𝒫 bunfold 
Set.powerset: {α : Type u_1} → Set α → Set (Set α)
Set.powerset
α✝: Type u_1

a: Set α✝

B: Set (Set α✝)

ha: a ∈ B

h₁: a ⊆ ⋃₀ B

h₂: 𝒫 a ⊆ 𝒫⋃₀ B

b: Set (Set α✝)

hb: 𝒫⋃₀ B = b

{t | t ⊆ b}
  
α✝: Type u_1

a: Set α✝

B: Set (Set α✝)

ha: a ∈ B

𝒫 a ∈ 𝒫 𝒫⋃₀ Brw [
α✝: Type u_1

a: Set α✝

B: Set (Set α✝)

ha: a ∈ B

h₁: a ⊆ ⋃₀ B

h₂: 𝒫 a ⊆ 𝒫⋃₀ B

b: Set (Set α✝)

hb: 𝒫⋃₀ B = b

𝒫 a ∈ {t | t ⊆ b}← 
hb: 𝒫⋃₀ B = b
hb,
α✝: Type u_1

a: Set α✝

B: Set (Set α✝)

ha: a ∈ B

h₁: a ⊆ ⋃₀ B

h₂: 𝒫 a ⊆ 𝒫⋃₀ B

b: Set (Set α✝)

hb: 𝒫⋃₀ B = b

𝒫 a ∈ {t | t ⊆ 𝒫⋃₀ B} 
α✝: Type u_1

a: Set α✝

B: Set (Set α✝)

ha: a ∈ B

h₁: a ⊆ ⋃₀ B

h₂: 𝒫 a ⊆ 𝒫⋃₀ B

b: Set (Set α✝)

hb: 𝒫⋃₀ B = b

𝒫 a ∈ {t | t ⊆ b}
Set.mem_setOf_eq: ∀ {α : Type ?u.15326} {x : α} {p : α → Prop}, (x ∈ {y | p y}) = p x
Set.mem_setOf_eq
α✝: Type u_1

a: Set α✝

B: Set (Set α✝)

ha: a ∈ B

h₁: a ⊆ ⋃₀ B

h₂: 𝒫 a ⊆ 𝒫⋃₀ B

b: Set (Set α✝)

hb: 𝒫⋃₀ B = b

𝒫 a ⊆ 𝒫⋃₀ B]
α✝: Type u_1

a: Set α✝

B: Set (Set α✝)

ha: a ∈ B

h₁: a ⊆ ⋃₀ B

h₂: 𝒫 a ⊆ 𝒫⋃₀ B

b: Set (Set α✝)

hb: 𝒫⋃₀ B = b

𝒫 a ⊆ 𝒫⋃₀ B
  
α✝: Type u_1

a: Set α✝

B: Set (Set α✝)

ha: a ∈ B

𝒫 a ∈ 𝒫 𝒫⋃₀ Bexact 
h₂: ?m.15201
h₂
Goals accomplished! 🐙

/-- ### Exercise 4.11 (i)

Show that for any sets `A` and `B`, `A = (A ∩ B) ∪ (A - B)`.
-/
theorem 
exercise_4_11_i: ∀ {α : Type u_1} {A B : Set α}, A = A ∩ B ∪ A \ B
exercise_4_11_i {
A: Set α
A 
B: Set α
B : 
Set: Type ?u.15460 → Type ?u.15460
Set 
α: ?m.15454
α}
  : 
A: Set α
A = (
A: Set α
A ∩ 
B: Set α
B) ∪ (
A: Set α
A \ 
B: Set α
B) := by
Goals accomplished! 🐙
  
α: Type u_1

A, B: Set α

A = A ∩ B ∪ A \ Bshow 
A: Set α
A = fun 
a: ?m.15530
a => 
A: Set α
A 
a: ?m.15530
a ∧ 
B: Set α
B 
a: ?m.15530
a ∨ 
A: Set α
A 
a: ?m.15530
a ∧ ¬
B: Set α
B 
a: ?m.15530
a
α: Type u_1

A, B: Set α

A = fun a => A a ∧ B a ∨ A a ∧ ¬B a
  
α: Type u_1

A, B: Set α

A = A ∩ B ∪ A \ Bsuffices ∀ 
x: ?m.15553
x, (
A: Set α
A 
x: ?m.15553
x ∧ (
B: Set α
B 
x: ?m.15553
x ∨ ¬
B: Set α
B 
x: ?m.15553
x)) = 
A: Set α
A 
x: ?m.15553
x by
    
α: Type u_1

A, B: Set α

A = fun a => A a ∧ B a ∨ A a ∧ ¬B aconv =>
α: Type u_1

A, B: Set α

this: ∀ (x : α), (A x ∧ (B x ∨ ¬B x)) = A x

x: α

h
A x 
α: Type u_1

A, B: Set α

this: ∀ (x : α), (A x ∧ (B x ∨ ¬B x)) = A x

A = fun a => A a ∧ B a ∨ A a ∧ ¬B arhs
α: Type u_1

A, B: Set α

this: ∀ (x : α), (A x ∧ (B x ∨ ¬B x)) = A x

fun a => A a ∧ B a ∨ A a ∧ ¬B a;
α: Type u_1

A, B: Set α

this: ∀ (x : α), (A x ∧ (B x ∨ ¬B x)) = A x

fun a => A a ∧ B a ∨ A a ∧ ¬B a 
α: Type u_1

A, B: Set α

this: ∀ (x : α), (A x ∧ (B x ∨ ¬B x)) = A x

A = fun a => A a ∧ B a ∨ A a ∧ ¬B aext x
α: Type u_1

A, B: Set α

this: ∀ (x : α), (A x ∧ (B x ∨ ¬B x)) = A x

x: α

h
A x ∧ B x ∨ A x ∧ ¬B x;
α: Type u_1

A, B: Set α

this: ∀ (x : α), (A x ∧ (B x ∨ ¬B x)) = A x

x: α

h
A x ∧ B x ∨ A x ∧ ¬B x 
α: Type u_1

A, B: Set α

this: ∀ (x : α), (A x ∧ (B x ∨ ¬B x)) = A x

A = fun a => A a ∧ B a ∨ A a ∧ ¬B arw [
α: Type u_1

A, B: Set α

this: ∀ (x : α), (A x ∧ (B x ∨ ¬B x)) = A x

x: α

h
A x ∧ B x ∨ A x ∧ ¬B x← 
and_or_left: ∀ {a b c : Prop}, a ∧ (b ∨ c) ↔ a ∧ b ∨ a ∧ c
and_or_left,
α: Type u_1

A, B: Set α

this: ∀ (x : α), (A x ∧ (B x ∨ ¬B x)) = A x

x: α

h
A x ∧ (B x ∨ ¬B x) 
α: Type u_1

A, B: Set α

this: ∀ (x : α), (A x ∧ (B x ∨ ¬B x)) = A x

x: α

h
A x ∧ B x ∨ A x ∧ ¬B x
this: ∀ (x : α), (A x ∧ (B x ∨ ¬B x)) = A x
this
α: Type u_1

A, B: Set α

this: ∀ (x : α), (A x ∧ (B x ∨ ¬B x)) = A x

x: α

h
A x]
α: Type u_1

A, B: Set α

this: ∀ (x : α), (A x ∧ (B x ∨ ¬B x)) = A x

x: α

h
A x
  
α: Type u_1

A, B: Set α

A = A ∩ B ∪ A \ Bintro 
x: α
x
α: Type u_1

A, B: Set α

x: α

(A x ∧ (B x ∨ ¬B x)) = A x
  
α: Type u_1

A, B: Set α

A = A ∩ B ∪ A \ Brefine 
propext: ∀ {a b : Prop}, (a ↔ b) → a = b
propext 
?_: ?m.15647 ↔ ?m.15648
?_
α: Type u_1

A, B: Set α

x: α

A x ∧ (B x ∨ ¬B x) ↔ A x
  
α: Type u_1

A, B: Set α

A = A ∩ B ∪ A \ Bapply 
Iff.intro: ∀ {a b : Prop}, (a → b) → (b → a) → (a ↔ b)
Iff.intro
α: Type u_1

A, B: Set α

x: α

mp
A x ∧ (B x ∨ ¬B x) → A x
α: Type u_1

A, B: Set α

x: α

mpr
A x → A x ∧ (B x ∨ ¬B x)
  
α: Type u_1

A, B: Set α

A = A ∩ B ∪ A \ B·
α: Type u_1

A, B: Set α

x: α

mp
A x ∧ (B x ∨ ¬B x) → A x 
α: Type u_1

A, B: Set α

x: α

mp
A x ∧ (B x ∨ ¬B x) → A x
α: Type u_1

A, B: Set α

x: α

mpr
A x → A x ∧ (B x ∨ ¬B x)intro 
hx: ?a
hx
α: Type u_1

A, B: Set α

x: α

hx: A x ∧ (B x ∨ ¬B x)

mp
A x
    
α: Type u_1

A, B: Set α

x: α

mp
A x ∧ (B x ∨ ¬B x) → A xexact 
hx: ?a
hx.
left: ∀ {a b : Prop}, a ∧ b → a
left
Goals accomplished! 🐙
  
α: Type u_1

A, B: Set α

A = A ∩ B ∪ A \ B·
α: Type u_1

A, B: Set α

x: α

mpr
A x → A x ∧ (B x ∨ ¬B x) 
α: Type u_1

A, B: Set α

x: α

mpr
A x → A x ∧ (B x ∨ ¬B x)intro 
hx: ?b
hx
α: Type u_1

A, B: Set α

x: α

hx: A x

mpr
A x ∧ (B x ∨ ¬B x)
    
α: Type u_1

A, B: Set α

x: α

mpr
A x → A x ∧ (B x ∨ ¬B x)exact ⟨
hx: ?b
hx, 
em: ∀ (p : Prop), p ∨ ¬p
em (
B: Set α
B 
x: α
x)⟩
Goals accomplished! 🐙

/-- ### Exercise 4.11 (ii)

Show that for any sets `A` and `B`, `A ∪ (B - A) = A ∪ B`.
-/
theorem 
exercise_4_11_ii: ∀ {α : Type u_1} {A B : Set α}, A ∪ B \ A = A ∪ B
exercise_4_11_ii {
A: Set α
A 
B: Set α
B : 
Set: Type ?u.15712 → Type ?u.15712
Set 
α: ?m.15709
α}
  : 
A: Set α
A ∪ (
B: Set α
B \ 
A: Set α
A) = 
A: Set α
A ∪ 
B: Set α
B := by
Goals accomplished! 🐙
  
α: Type u_1

A, B: Set α

A ∪ B \ A = A ∪ Bshow (fun 
a: ?m.15774
a => 
A: Set α
A 
a: ?m.15774
a ∨ 
B: Set α
B 
a: ?m.15774
a ∧ ¬
A: Set α
A 
a: ?m.15774
a) = fun 
a: ?m.15777
a => 
A: Set α
A 
a: ?m.15777
a ∨ 
B: Set α
B 
a: ?m.15777
a
α: Type u_1

A, B: Set α

(fun a => A a ∨ B a ∧ ¬A a) = fun a => A a ∨ B a
  
α: Type u_1

A, B: Set α

A ∪ B \ A = A ∪ Bsuffices ∀ 
x: ?m.15806
x, ((
A: Set α
A 
x: ?m.15806
x ∨ 
B: Set α
B 
x: ?m.15806
x) ∧ (
A: Set α
A 
x: ?m.15806
x ∨ ¬
A: Set α
A 
x: ?m.15806
x)) = (
A: Set α
A 
x: ?m.15806
x ∨ 
B: Set α
B 
x: ?m.15806
x) by
    
α: Type u_1

A, B: Set α

(fun a => A a ∨ B a ∧ ¬A a) = fun a => A a ∨ B aconv =>
α: Type u_1

A, B: Set α

this: ∀ (x : α), ((A x ∨ B x) ∧ (A x ∨ ¬A x)) = (A x ∨ B x)

x: α

h
A x ∨ B x 
α: Type u_1

A, B: Set α

this: ∀ (x : α), ((A x ∨ B x) ∧ (A x ∨ ¬A x)) = (A x ∨ B x)

(fun a => A a ∨ B a ∧ ¬A a) = fun a => A a ∨ B alhs
α: Type u_1

A, B: Set α

this: ∀ (x : α), ((A x ∨ B x) ∧ (A x ∨ ¬A x)) = (A x ∨ B x)

fun a => A a ∨ B a ∧ ¬A a;
α: Type u_1

A, B: Set α

this: ∀ (x : α), ((A x ∨ B x) ∧ (A x ∨ ¬A x)) = (A x ∨ B x)

fun a => A a ∨ B a ∧ ¬A a 
α: Type u_1

A, B: Set α

this: ∀ (x : α), ((A x ∨ B x) ∧ (A x ∨ ¬A x)) = (A x ∨ B x)

(fun a => A a ∨ B a ∧ ¬A a) = fun a => A a ∨ B aext x
α: Type u_1

A, B: Set α

this: ∀ (x : α), ((A x ∨ B x) ∧ (A x ∨ ¬A x)) = (A x ∨ B x)

x: α

h
A x ∨ B x ∧ ¬A x;
α: Type u_1

A, B: Set α

this: ∀ (x : α), ((A x ∨ B x) ∧ (A x ∨ ¬A x)) = (A x ∨ B x)

x: α

h
A x ∨ B x ∧ ¬A x 
α: Type u_1

A, B: Set α

this: ∀ (x : α), ((A x ∨ B x) ∧ (A x ∨ ¬A x)) = (A x ∨ B x)

(fun a => A a ∨ B a ∧ ¬A a) = fun a => A a ∨ B arw [
α: Type u_1

A, B: Set α

this: ∀ (x : α), ((A x ∨ B x) ∧ (A x ∨ ¬A x)) = (A x ∨ B x)

x: α

h
A x ∨ B x ∧ ¬A x
or_and_left: ∀ {a b c : Prop}, a ∨ b ∧ c ↔ (a ∨ b) ∧ (a ∨ c)
or_and_left,
α: Type u_1

A, B: Set α

this: ∀ (x : α), ((A x ∨ B x) ∧ (A x ∨ ¬A x)) = (A x ∨ B x)

x: α

h
(A x ∨ B x) ∧ (A x ∨ ¬A x) 
α: Type u_1

A, B: Set α

this: ∀ (x : α), ((A x ∨ B x) ∧ (A x ∨ ¬A x)) = (A x ∨ B x)

x: α

h
A x ∨ B x ∧ ¬A x
this: ∀ (x : α), ((A x ∨ B x) ∧ (A x ∨ ¬A x)) = (A x ∨ B x)
this 
x: ?α
x
α: Type u_1

A, B: Set α

this: ∀ (x : α), ((A x ∨ B x) ∧ (A x ∨ ¬A x)) = (A x ∨ B x)

x: α

h
A x ∨ B x]
α: Type u_1

A, B: Set α

this: ∀ (x : α), ((A x ∨ B x) ∧ (A x ∨ ¬A x)) = (A x ∨ B x)

x: α

h
A x ∨ B x
  
α: Type u_1

A, B: Set α

A ∪ B \ A = A ∪ Bintro 
x: α
x
α: Type u_1

A, B: Set α

x: α

((A x ∨ B x) ∧ (A x ∨ ¬A x)) = (A x ∨ B x)
  
α: Type u_1

A, B: Set α

A ∪ B \ A = A ∪ Brefine 
propext: ∀ {a b : Prop}, (a ↔ b) → a = b
propext 
?_: ?m.15897 ↔ ?m.15898
?_
α: Type u_1

A, B: Set α

x: α

(A x ∨ B x) ∧ (A x ∨ ¬A x) ↔ A x ∨ B x
  
α: Type u_1

A, B: Set α

A ∪ B \ A = A ∪ Bapply 
Iff.intro: ∀ {a b : Prop}, (a → b) → (b → a) → (a ↔ b)
Iff.intro
α: Type u_1

A, B: Set α

x: α

mp
(A x ∨ B x) ∧ (A x ∨ ¬A x) → A x ∨ B x
α: Type u_1

A, B: Set α

x: α

mpr
A x ∨ B x → (A x ∨ B x) ∧ (A x ∨ ¬A x)
  
α: Type u_1

A, B: Set α

A ∪ B \ A = A ∪ B·
α: Type u_1

A, B: Set α

x: α

mp
(A x ∨ B x) ∧ (A x ∨ ¬A x) → A x ∨ B x 
α: Type u_1

A, B: Set α

x: α

mp
(A x ∨ B x) ∧ (A x ∨ ¬A x) → A x ∨ B x
α: Type u_1

A, B: Set α

x: α

mpr
A x ∨ B x → (A x ∨ B x) ∧ (A x ∨ ¬A x)intro 
hx: ?a
hx
α: Type u_1

A, B: Set α

x: α

hx: (A x ∨ B x) ∧ (A x ∨ ¬A x)

mp
A x ∨ B x
    
α: Type u_1

A, B: Set α

x: α

mp
(A x ∨ B x) ∧ (A x ∨ ¬A x) → A x ∨ B xexact 
hx: ?a
hx.
left: ∀ {a b : Prop}, a ∧ b → a
left
Goals accomplished! 🐙
  
α: Type u_1

A, B: Set α

A ∪ B \ A = A ∪ B·
α: Type u_1

A, B: Set α

x: α

mpr
A x ∨ B x → (A x ∨ B x) ∧ (A x ∨ ¬A x) 
α: Type u_1

A, B: Set α

x: α

mpr
A x ∨ B x → (A x ∨ B x) ∧ (A x ∨ ¬A x)intro 
hx: ?b
hx
α: Type u_1

A, B: Set α

x: α

hx: A x ∨ B x

mpr
(A x ∨ B x) ∧ (A x ∨ ¬A x)
    
α: Type u_1

A, B: Set α

x: α

mpr
A x ∨ B x → (A x ∨ B x) ∧ (A x ∨ ¬A x)exact ⟨
hx: ?b
hx, 
em: ∀ (p : Prop), p ∨ ¬p
em (
A: Set α
A 
x: α
x)⟩
Goals accomplished! 🐙

section

variable {
A: Set ℕ
A 
B: Set ℕ
B 
C: Set ℕ
C : 
Set: Type ?u.20689 → Type ?u.20689
Set 
ℕ: Type
ℕ}

variable {
hA: A = {1, 2, 3}
hA : 
A: Set ℕ
A = {
1: ?m.16964
1, 
2: ?m.16008
2, 
3: ?m.16029
3}}
variable {
hB: B = {2, 3, 4}
hB : 
B: Set ℕ
B = {
2: ?m.16331
2, 
3: ?m.16352
3, 
4: ?m.16373
4}}
variable {
hC: C = {3, 4, 5}
hC : 
C: Set ℕ
C = {
3: ?m.17281
3, 
4: ?m.16827
4, 
5: ?m.16848
5}}

lemma 
right_diff_eq_insert_one_three: A \ (B \ C) = {1, 3}
right_diff_eq_insert_one_three : 
A: Set ℕ
A \ (
B: Set ℕ
B \ 
C: Set ℕ
C) = {
1: ?m.17463
1, 
3: ?m.17485
3} := by
Goals accomplished! 🐙
  
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

A \ (B \ C) = {1, 3}rw [
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

A \ (B \ C) = {1, 3}
hA: A = {1, 2, 3}
hA,
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

{1, 2, 3} \ (B \ C) = {1, 3} 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

A \ (B \ C) = {1, 3}
hB: B = {2, 3, 4}
hB,
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

{1, 2, 3} \ ({2, 3, 4} \ C) = {1, 3} 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

A \ (B \ C) = {1, 3}
hC: C = {3, 4, 5}
hC
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

{1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5}) = {1, 3}]
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

{1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5}) = {1, 3}
  
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

A \ (B \ C) = {1, 3}ext 
x: ?α
x
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h
x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5}) ↔ x ∈ {1, 3}
  
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

A \ (B \ C) = {1, 3}apply 
Iff.intro: ∀ {a b : Prop}, (a → b) → (b → a) → (a ↔ b)
Iff.intro
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5}) → x ∈ {1, 3}
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mpr
x ∈ {1, 3} → x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5})
  
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

A \ (B \ C) = {1, 3}·
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5}) → x ∈ {1, 3} 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5}) → x ∈ {1, 3}
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mpr
x ∈ {1, 3} → x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5})intro 
hx: ?a
hx
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5})

h.mp
x ∈ {1, 3}
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5}) → x ∈ {1, 3}unfold 
SDiff.sdiff: {α : Type u} → [self : SDiff α] → α → α → α
SDiff.sdiff 
Set.instSDiffSet: {α : Type u_1} → SDiff (Set α)
Set.instSDiffSet 
Set.diff: {α : Type u_1} → Set α → Set α → Set α
Set.diff at 
hx: x ∈ { sdiff := Set.diff }.1 {1, 2, 3} ({ sdiff := Set.diff }.1 {2, 3, 4} {3, 4, 5})
hx
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈   { sdiff := fun s t => {a | a ∈ s ∧ ¬a ∈ t} }.1 {1, 2, 3}
    ({ sdiff := fun s t => {a | a ∈ s ∧ ¬a ∈ t} }.1 {2, 3, 4} {3, 4, 5})

h.mp
x ∈ {1, 3}
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5}) → x ∈ {1, 3}unfold 
Membership.mem: {α : outParam (Type u)} → {γ : Type v} → [self : Membership α γ] → α → γ → Prop
Membership.mem 
Set.instMembershipSet: {α : Type u_1} → Membership α (Set α)
Set.instMembershipSet 
Set.Mem: {α : Type u_1} → α → Set α → Prop
Set.Mem 
setOf: {α : Type u} → (α → Prop) → Set α
setOf at 
hx: x ∈   { sdiff := fun s t => {a | a ∈ s ∧ ¬a ∈ t} }.1 {1, 2, 3}
    ({ sdiff := fun s t => {a | a ∈ s ∧ ¬a ∈ t} }.1 {2, 3, 4} {3, 4, 5})
hx
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1 {1, 2, 3}
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1 {2, 3, 4}
      {3, 4, 5}))

h.mp
x ∈ {1, 3}
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5}) → x ∈ {1, 3}unfold 
insert: {α : outParam (Type u)} → {γ : Type v} → [self : Insert α γ] → α → γ → γ
insert 
Set.instInsertSet: {α : Type u_1} → Insert α (Set α)
Set.instInsertSet 
Set.insert: {α : Type u_1} → α → Set α → Set α
Set.insert 
setOf: {α : Type u} → (α → Prop) → Set α
setOf at 
hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := Set.insert }.1 1 ({ insert := Set.insert }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := Set.insert }.1 2 ({ insert := Set.insert }.1 3 {4}))
      ({ insert := Set.insert }.1 3 ({ insert := Set.insert }.1 4 {5}))))
hx
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

h.mp
x ∈ {1, 3}
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5}) → x ∈ {1, 3}have ⟨
ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
ha, 
hb: ¬{ mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))
hb⟩ := 
hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))
hx
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬{ mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

h.mp
x ∈ {1, 3}
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5}) → x ∈ {1, 3}rw [
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬{ mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

h.mp
x ∈ {1, 3}
not_and_de_morgan: ∀ {p q : Prop}, ¬(p ∧ q) ↔ ¬p ∨ ¬q
not_and_de_morgan,
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬{ mem := fun a s => s a }.1 x
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})) ∨   ¬¬{ mem := fun a s => s a }.1 x
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))

h.mp
x ∈ {1, 3} 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬{ mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

h.mp
x ∈ {1, 3}
not_or_de_morgan: ∀ {p q : Prop}, ¬(p ∨ q) ↔ ¬p ∧ ¬q
not_or_de_morgan
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4} ∨   ¬¬{ mem := fun a s => s a }.1 x
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))

h.mp
x ∈ {1, 3}]
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4} ∨   ¬¬{ mem := fun a s => s a }.1 x
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))

h.mp
x ∈ {1, 3} at 
hb: ¬{ mem := fun a s => s a }.1 x
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})) ∨   ¬¬{ mem := fun a s => s a }.1 x
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))
hb
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4} ∨   ¬¬{ mem := fun a s => s a }.1 x
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))

h.mp
x ∈ {1, 3}
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5}) → x ∈ {1, 3}simp only [
Set.mem_singleton_iff: ∀ {α : Type ?u.17914} {a b : α}, a ∈ {b} ↔ a = b
Set.mem_singleton_iff, 
not_not: ∀ {a : Prop}, ¬¬a ↔ a
not_not] at 
hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4} ∨   ¬¬{ mem := fun a s => s a }.1 x
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))
hb
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

h.mp
x ∈ {1, 3}
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5}) → x ∈ {1, 3}refine 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
ha 
Or.inl: ∀ {a b : Prop}, a → a ∨ b
Or.inl 
?_: ?m.18264 → ?m.18265
?_
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

h.mp
x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3} → x ∈ {1, 3}
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5}) → x ∈ {1, 3}intro 
hy: ?m.18264
hy
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp
x ∈ {1, 3}
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5}) → x ∈ {1, 3}apply 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5
hb
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.left
(¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) → x ∈ {1, 3}
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.right
(x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5) → x ∈ {1, 3}
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5}) → x ∈ {1, 3}·
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.left
(¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) → x ∈ {1, 3} 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.left
(¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) → x ∈ {1, 3}
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.right
(x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5) → x ∈ {1, 3}intro 
hz: ¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4
hz
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: ¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4

h.mp.left
x ∈ {1, 3}
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.left
(¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) → x ∈ {1, 3}exact 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
hy: ?m.18264
hy (
absurd: ∀ {a : Prop} {b : Sort ?u.18293}, a → ¬a → b
absurd · 
hz: ¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4
hz.
left: ∀ {a b : Prop}, a ∧ b → a
left) 
Or.inr: ∀ {a b : Prop}, b → a ∨ b
Or.inr
Goals accomplished! 🐙
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5}) → x ∈ {1, 3}·
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.right
(x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5) → x ∈ {1, 3} 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.right
(x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5) → x ∈ {1, 3}intro 
hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5
hz
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

h.mp.right
x ∈ {1, 3}
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.right
(x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5) → x ∈ {1, 3}refine 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5
hz 
Or.inr: ∀ {a b : Prop}, b → a ∨ b
Or.inr 
?_: ?m.18310 → ?m.18311
?_
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

h.mp.right
(x ∈ fun b => b = 4 ∨ b = 5) → x ∈ {1, 3}
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.right
(x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5) → x ∈ {1, 3}intro 
hz₁: ?m.18310
hz₁
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

h.mp.right
x ∈ {1, 3}
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.right
(x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5) → x ∈ {1, 3}apply 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
hy: ?m.18264
hy
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

h.mp.right.left
x = 2 → x ∈ {1, 3}
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

h.mp.right.right
x ∈ {3} → x ∈ {1, 3} 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

h.mp.right
x ∈ {1, 3}<;>
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

h.mp.right.left
x = 2 → x ∈ {1, 3}
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

h.mp.right.right
x ∈ {3} → x ∈ {1, 3} 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

h.mp.right
x ∈ {1, 3}apply 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
hz₁: ?m.18310
hz₁
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

h.mp.right.left.left
x = 4 → x = 2 → x ∈ {1, 3}
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

h.mp.right.left.right
x = 5 → x = 2 → x ∈ {1, 3} 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

h.mp.right
x ∈ {1, 3}<;>
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

h.mp.right.left.left
x = 4 → x = 2 → x ∈ {1, 3}
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

h.mp.right.left.right
x = 5 → x = 2 → x ∈ {1, 3}
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

h.mp.right.right.left
x = 4 → x ∈ {3} → x ∈ {1, 3}
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

h.mp.right.right.right
x = 5 → x ∈ {3} → x ∈ {1, 3}
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

h.mp.right
x ∈ {1, 3}·
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

h.mp.right.left.right
x = 5 → x = 2 → x ∈ {1, 3} 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

h.mp.right.left.right
x = 5 → x = 2 → x ∈ {1, 3}intro 
hz₂: x = 5
hz₂ 
hz₃: x = 2
hz₃
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

hz₂: x = 5

hz₃: x ∈ {3}

h.mp.right.right.right
x ∈ {1, 3}
        
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

h.mp.right.left.left
x = 4 → x = 2 → x ∈ {1, 3}rw [
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

hz₂: x = 4

hz₃: x ∈ {3}

h.mp.right.right.left
x ∈ {1, 3}
hz₂: x = 5
hz₂
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

hz₂: x = 4

hz₃: 4 ∈ {3}

h.mp.right.right.left
x ∈ {1, 3}]
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

hz₂: x = 5

hz₃: 5 = 2

h.mp.right.left.right
x ∈ {1, 3} at 
hz₃: x ∈ {3}
hz₃
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

hz₂: x = 4

hz₃: 4 ∈ {3}

h.mp.right.right.left
x ∈ {1, 3}
        
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: (¬x = 2 ∧ ¬x ∈ fun b => b = 3 ∨ b = 4) ∨ x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 3 ∨ x ∈ fun b => b = 4 ∨ b = 5

hz₁: x ∈ fun b => b = 4 ∨ b = 5

h.mp.right.left.left
x = 4 → x = 2 → x ∈ {1, 3}simp at 
hz₃: 4 = 2
hz₃
Goals accomplished! 🐙
  
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

A \ (B \ C) = {1, 3}·
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mpr
x ∈ {1, 3} → x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5}) 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mpr
x ∈ {1, 3} → x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5})intro 
hx: ?b
hx
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr
x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5})
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mpr
x ∈ {1, 3} → x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5})unfold 
SDiff.sdiff: {α : Type u} → [self : SDiff α] → α → α → α
SDiff.sdiff 
Set.instSDiffSet: {α : Type u_1} → SDiff (Set α)
Set.instSDiffSet 
Set.diff: {α : Type u_1} → Set α → Set α → Set α
Set.diff
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr
x ∈   { sdiff := fun s t => {a | a ∈ s ∧ ¬a ∈ t} }.1 {1, 2, 3}
    ({ sdiff := fun s t => {a | a ∈ s ∧ ¬a ∈ t} }.1 {2, 3, 4} {3, 4, 5})
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mpr
x ∈ {1, 3} → x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5})unfold 
Membership.mem: {α : outParam (Type u)} → {γ : Type v} → [self : Membership α γ] → α → γ → Prop
Membership.mem 
Set.instMembershipSet: {α : Type u_1} → Membership α (Set α)
Set.instMembershipSet 
Set.Mem: {α : Type u_1} → α → Set α → Prop
Set.Mem 
setOf: {α : Type u} → (α → Prop) → Set α
setOf
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr
{ mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1 {1, 2, 3}
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1 {2, 3, 4}
      {3, 4, 5}))
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mpr
x ∈ {1, 3} → x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5})unfold 
insert: {α : outParam (Type u)} → {γ : Type v} → [self : Insert α γ] → α → γ → γ
insert 
Set.instInsertSet: {α : Type u_1} → Insert α (Set α)
Set.instInsertSet 
Set.insert: {α : Type u_1} → α → Set α → Set α
Set.insert 
setOf: {α : Type u} → (α → Prop) → Set α
setOf
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr
{ mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mpr
x ∈ {1, 3} → x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5})apply 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
hx: ?b
hx
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr.left
x = 1 →
  { mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr.right
x ∈ {3} →
  { mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mpr
x ∈ {1, 3} → x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5})·
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr.left
x = 1 →
  { mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))) 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr.left
x = 1 →
  { mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr.right
x ∈ {3} →
  { mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))intro 
hy: x = 1
hy
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

hy: x = 1

h.mpr.left
{ mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr.left
x = 1 →
  { mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))refine ⟨
Or.inl: ∀ {a b : Prop}, a → a ∨ b
Or.inl 
hy: x = 1
hy, 
?_: ?m.18582
?_⟩
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

hy: x = 1

h.mpr.left
¬{ mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr.left
x = 1 →
  { mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))intro 
hz: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))
hz
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

hy: x = 1

hz: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

h.mpr.left
False
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr.left
x = 1 →
  { mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))rw [
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

hy: x = 1

hz: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

h.mpr.left
False
hy: x = 1
hy
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

hy: x = 1

hz: { mem := fun a s => s a }.1 1
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

h.mpr.left
False]
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

hy: x = 1

hz: { mem := fun a s => s a }.1 1
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

h.mpr.left
False at 
hz: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))
hz
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

hy: x = 1

hz: { mem := fun a s => s a }.1 1
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

h.mpr.left
False
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr.left
x = 1 →
  { mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))unfold 
Membership.mem: {α : outParam (Type u)} → {γ : Type v} → [self : Membership α γ] → α → γ → Prop
Membership.mem 
Set.instMembershipSet: {α : Type u_1} → Membership α (Set α)
Set.instMembershipSet 
Set.Mem: {α : Type u_1} → α → Set α → Prop
Set.Mem at 
hz: { mem := fun a s => s a }.1 1
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ Set.instMembershipSet.1 b s }.1 2
      ({ insert := fun a s b => b = a ∨ Set.instMembershipSet.1 b s }.1 3 {4}))
    ({ insert := fun a s b => b = a ∨ Set.instMembershipSet.1 b s }.1 3
      ({ insert := fun a s b => b = a ∨ Set.instMembershipSet.1 b s }.1 4 {5})))
hz
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

hy: x = 1

hz: { mem := fun a s => s a }.1 1
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ { mem := fun a s => s a }.1 b s }.1 2
      ({ insert := fun a s b => b = a ∨ { mem := fun a s => s a }.1 b s }.1 3 {4}))
    ({ insert := fun a s b => b = a ∨ { mem := fun a s => s a }.1 b s }.1 3
      ({ insert := fun a s b => b = a ∨ { mem := fun a s => s a }.1 b s }.1 4 {5})))

h.mpr.left
False
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr.left
x = 1 →
  { mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))unfold 
singleton: {α : outParam (Type u)} → {β : Type v} → [self : Singleton α β] → α → β
singleton 
Set.instSingletonSet: {α : Type u_1} → Singleton α (Set α)
Set.instSingletonSet 
Set.singleton: {α : Type u_1} → α → Set α
Set.singleton 
setOf: {α : Type u} → (α → Prop) → Set α
setOf at 
hz: { mem := fun a s => s a }.1 1
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ { mem := fun a s => s a }.1 b s }.1 2
      ({ insert := fun a s b => b = a ∨ { mem := fun a s => s a }.1 b s }.1 3 {4}))
    ({ insert := fun a s b => b = a ∨ { mem := fun a s => s a }.1 b s }.1 3
      ({ insert := fun a s b => b = a ∨ { mem := fun a s => s a }.1 b s }.1 4 {5})))
hz
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

hy: x = 1

hz: { mem := fun a s => s a }.1 1
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ { mem := fun a s => s a }.1 b s }.1 2
      ({ insert := fun a s b => b = a ∨ { mem := fun a s => s a }.1 b s }.1 3 ({ singleton := fun a b => b = a }.1 4)))
    ({ insert := fun a s b => b = a ∨ { mem := fun a s => s a }.1 b s }.1 3
      ({ insert := fun a s b => b = a ∨ { mem := fun a s => s a }.1 b s }.1 4 ({ singleton := fun a b => b = a }.1 5))))

h.mpr.left
False
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr.left
x = 1 →
  { mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))simp only at 
hz: { mem := fun a s => s a }.1 1
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ { mem := fun a s => s a }.1 b s }.1 2
      ({ insert := fun a s b => b = a ∨ { mem := fun a s => s a }.1 b s }.1 3 ({ singleton := fun a b => b = a }.1 4)))
    ({ insert := fun a s b => b = a ∨ { mem := fun a s => s a }.1 b s }.1 3
      ({ insert := fun a s b => b = a ∨ { mem := fun a s => s a }.1 b s }.1 4 ({ singleton := fun a b => b = a }.1 5))))
hz
Goals accomplished! 🐙
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mpr
x ∈ {1, 3} → x ∈ {1, 2, 3} \ ({2, 3, 4} \ {3, 4, 5})·
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr.right
x ∈ {3} →
  { mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))) 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr.right
x ∈ {3} →
  { mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))intro 
hy: x ∈ {3}
hy
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

hy: x ∈ {3}

h.mpr.right
{ mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr.right
x ∈ {3} →
  { mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))refine ⟨
Or.inr: ∀ {a b : Prop}, b → a ∨ b
Or.inr (
Or.inr: ∀ {a b : Prop}, b → a ∨ b
Or.inr 
hy: x ∈ {3}
hy), 
?_: ?m.18744
?_⟩
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

hy: x ∈ {3}

h.mpr.right
¬{ mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr.right
x ∈ {3} →
  { mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))intro 
hz: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))
hz
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

hy: x ∈ {3}

hz: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

h.mpr.right
False
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr.right
x ∈ {3} →
  { mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))have 
hzr: ?m.18754
hzr := 
hz: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))
hz.
right: ∀ {a b : Prop}, a ∧ b → b
right
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

hy: x ∈ {3}

hz: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

hzr: ¬{ mem := fun a s => s a }.1 x
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))

h.mpr.right
False
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr.right
x ∈ {3} →
  { mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))rw [
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

hy: x ∈ {3}

hz: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

hzr: ¬{ mem := fun a s => s a }.1 x
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))

h.mpr.right
False
not_or_de_morgan: ∀ {p q : Prop}, ¬(p ∨ q) ↔ ¬p ∧ ¬q
not_or_de_morgan
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

hy: x ∈ {3}

hz: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

hzr: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

h.mpr.right
False]
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

hy: x ∈ {3}

hz: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

hzr: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

h.mpr.right
False at 
hzr: ?m.18754
hzr
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

hy: x ∈ {3}

hz: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

hzr: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

h.mpr.right
False
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1, 3}

h.mpr.right
x ∈ {3} →
  { mem := fun a s => s a }.1 x
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
        ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))))exact 
absurd: ∀ {a : Prop} {b : Sort ?u.18788}, a → ¬a → b
absurd 
hy: x ∈ {3}
hy 
hzr: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}
hzr.
left: ∀ {a b : Prop}, a ∧ b → a
left
Goals accomplished! 🐙

lemma 
left_diff_eq_singleton_one: (A \ B) \ C = {1}
left_diff_eq_singleton_one : (
A: Set ℕ
A \ 
B: Set ℕ
B) \ 
C: Set ℕ
C = {
1: ?m.19394
1} := by
Goals accomplished! 🐙
  
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

(A \ B) \ C = {1}rw [
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

(A \ B) \ C = {1}
hA: A = {1, 2, 3}
hA,
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

({1, 2, 3} \ B) \ C = {1} 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

(A \ B) \ C = {1}
hB: B = {2, 3, 4}
hB,
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

({1, 2, 3} \ {2, 3, 4}) \ C = {1} 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

(A \ B) \ C = {1}
hC: C = {3, 4, 5}
hC
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5} = {1}]
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5} = {1}
  
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

(A \ B) \ C = {1}ext 
x: ?α
x
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h
x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5} ↔ x ∈ {1}
  
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

(A \ B) \ C = {1}apply 
Iff.intro: ∀ {a b : Prop}, (a → b) → (b → a) → (a ↔ b)
Iff.intro
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5} → x ∈ {1}
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mpr
x ∈ {1} → x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5}
  
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

(A \ B) \ C = {1}·
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5} → x ∈ {1} 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5} → x ∈ {1}
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mpr
x ∈ {1} → x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5}intro 
hx: ?a
hx
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5}

h.mp
x ∈ {1}
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5} → x ∈ {1}unfold 
SDiff.sdiff: {α : Type u} → [self : SDiff α] → α → α → α
SDiff.sdiff 
Set.instSDiffSet: {α : Type u_1} → SDiff (Set α)
Set.instSDiffSet 
Set.diff: {α : Type u_1} → Set α → Set α → Set α
Set.diff at 
hx: ?a
hx
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈   { sdiff := fun s t => {a | a ∈ s ∧ ¬a ∈ t} }.1 ({ sdiff := fun s t => {a | a ∈ s ∧ ¬a ∈ t} }.1 {1, 2, 3} {2, 3, 4})
    {3, 4, 5}

h.mp
x ∈ {1}
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5} → x ∈ {1}unfold 
Membership.mem: {α : outParam (Type u)} → {γ : Type v} → [self : Membership α γ] → α → γ → Prop
Membership.mem 
Set.instMembershipSet: {α : Type u_1} → Membership α (Set α)
Set.instMembershipSet 
Set.Mem: {α : Type u_1} → α → Set α → Prop
Set.Mem 
setOf: {α : Type u} → (α → Prop) → Set α
setOf at 
hx: x ∈   { sdiff := fun s t => {a | a ∈ s ∧ ¬a ∈ t} }.1 ({ sdiff := fun s t => {a | a ∈ s ∧ ¬a ∈ t} }.1 {1, 2, 3} {2, 3, 4})
    {3, 4, 5}
hx
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1 {1, 2, 3} {2, 3, 4})
    {3, 4, 5})

h.mp
x ∈ {1}
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5} → x ∈ {1}unfold 
insert: {α : outParam (Type u)} → {γ : Type v} → [self : Insert α γ] → α → γ → γ
insert 
Set.instInsertSet: {α : Type u_1} → Insert α (Set α)
Set.instInsertSet 
Set.insert: {α : Type u_1} → α → Set α → Set α
Set.insert 
setOf: {α : Type u} → (α → Prop) → Set α
setOf at 
hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1 {1, 2, 3} {2, 3, 4})
    {3, 4, 5})
hx
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

h.mp
x ∈ {1}
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5} → x ∈ {1}have ⟨⟨
ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
ha, 
hb: ¬{ mem := fun a s => s a }.1 x
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
hb⟩, 
hc: ¬{ mem := fun a s => s a }.1 x
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))
hc⟩ := 
hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))
hx
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬{ mem := fun a s => s a }.1 x
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))

hc: ¬{ mem := fun a s => s a }.1 x
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))

h.mp
x ∈ {1}
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5} → x ∈ {1}rw [
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬{ mem := fun a s => s a }.1 x
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))

hc: ¬{ mem := fun a s => s a }.1 x
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))

h.mp
x ∈ {1}
not_or_de_morgan: ∀ {p q : Prop}, ¬(p ∨ q) ↔ ¬p ∧ ¬q
not_or_de_morgan
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

h.mp
x ∈ {1}]
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

h.mp
x ∈ {1} at 
hb: ¬{ mem := fun a s => s a }.1 x
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}))
hb 
hc: ¬{ mem := fun a s => s a }.1 x
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}))
hc
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

h.mp
x ∈ {1}
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5} → x ∈ {1}apply 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
ha
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

h.mp.left
x = 1 → x ∈ {1}
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

h.mp.right
x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3} → x ∈ {1}
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5} → x ∈ {1}·
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

h.mp.left
x = 1 → x ∈ {1} 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

h.mp.left
x = 1 → x ∈ {1}
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

h.mp.right
x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3} → x ∈ {1}simp
Goals accomplished! 🐙 
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mp
x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5} → x ∈ {1}·
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

h.mp.right
x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3} → x ∈ {1} 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

h.mp.right
x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3} → x ∈ {1}intro 
hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}
hy
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.right
x ∈ {1}
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

h.mp.right
x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3} → x ∈ {1}apply 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}
hy
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.right.left
x = 2 → x ∈ {1}
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.right.right
x ∈ {3} → x ∈ {1}
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

h.mp.right
x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3} → x ∈ {1}·
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.right.left
x = 2 → x ∈ {1} 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.right.left
x = 2 → x ∈ {1}
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.right.right
x ∈ {3} → x ∈ {1}intro 
hz: x = 2
hz
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x = 2

h.mp.right.left
x ∈ {1}
        
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.right.left
x = 2 → x ∈ {1}exact 
absurd: ∀ {a : Prop} {b : Sort ?u.20113}, a → ¬a → b
absurd 
hz: x = 2
hz 
hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}
hb.
left: ∀ {a b : Prop}, a ∧ b → a
left
Goals accomplished! 🐙
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

h.mp.right
x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3} → x ∈ {1}·
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.right.right
x ∈ {3} → x ∈ {1} 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.right.right
x ∈ {3} → x ∈ {1}intro 
hz: x ∈ {3}
hz
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

hz: x ∈ {3}

h.mp.right.right
x ∈ {1}
        
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: { mem := fun a s => s a }.1 x
  ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
    ({ sdiff := fun s t a => { mem := fun a s => s a }.1 a s ∧ ¬{ mem := fun a s => s a }.1 a t }.1
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))
      ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4})))
    ({ insert := fun a s b => b = a ∨ b ∈ s }.1 3 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5})))

ha: { mem := fun a s => s a }.1 x
  ({ insert := fun a s b => b = a ∨ b ∈ s }.1 1 ({ insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}))

hb: ¬x = 2 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 3 {4}

hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}

hy: x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 2 {3}

h.mp.right.right
x ∈ {3} → x ∈ {1}exact 
absurd: ∀ {a : Prop} {b : Sort ?u.20125}, a → ¬a → b
absurd 
hz: x ∈ {3}
hz 
hc: ¬x = 3 ∧ ¬x ∈ { insert := fun a s b => b = a ∨ b ∈ s }.1 4 {5}
hc.
left: ∀ {a b : Prop}, a ∧ b → a
left
Goals accomplished! 🐙
  
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

(A \ B) \ C = {1}·
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mpr
x ∈ {1} → x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5} 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mpr
x ∈ {1} → x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5}intro 
hx: ?b
hx
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

h.mpr
x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5}
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

h.mpr
x ∈ {1} → x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5}refine ⟨⟨
Or.inl: ∀ {a b : Prop}, a → a ∨ b
Or.inl 
hx: ?b
hx, 
?_: ?m.20145
?_⟩, 
?_: ?m.20139
?_⟩
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

h.mpr.refine_1
¬x ∈ {2, 3, 4}
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

h.mpr.refine_2
¬x ∈ {3, 4, 5} 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

h.mpr
x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5}<;>
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

h.mpr.refine_1
¬x ∈ {2, 3, 4}
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

h.mpr.refine_2
¬x ∈ {3, 4, 5}
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

h.mpr
x ∈ ({1, 2, 3} \ {2, 3, 4}) \ {3, 4, 5}·
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

h.mpr.refine_2
¬x ∈ {3, 4, 5} 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

h.mpr.refine_2
¬x ∈ {3, 4, 5}intro 
hy: x ∈ {3, 4, 5}
hy
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

hy: x ∈ {3, 4, 5}

h.mpr.refine_2
False
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

h.mpr.refine_2
¬x ∈ {3, 4, 5}cases 
hy: x ∈ {2, 3, 4}
hy with
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

hy: x ∈ {2, 3, 4}

h.mpr.refine_1
False| 
inl: ∀ {a b : Prop}, a → a ∨ b
inl 
y: ?m.20167
y => 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: x = 3

h.mpr.refine_2.inl
Falserw [
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: x = 2

h.mpr.refine_1.inl
False
hx: ?b
hx
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: 1 = 3

h.mpr.refine_2.inl
False]
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: 1 = 2

h.mpr.refine_1.inl
False at 
y: ?m.20167
y
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: 1 = 2

h.mpr.refine_1.inl
False;
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: 1 = 3

h.mpr.refine_2.inl
False 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: x = 2

h.mpr.refine_1.inl
Falsesimp at 
y: 1 = 2
y
Goals accomplished! 🐙
      
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

hy: x ∈ {2, 3, 4}

h.mpr.refine_1
False| 
inr: ∀ {a b : Prop}, b → a ∨ b
inr 
hz: ?m.20168
hz => 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

hz: x ∈ {4, 5}

h.mpr.refine_2.inr
Falsecases 
hz: ?m.20417
hz with
        
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

hz: x ∈ {3, 4}

h.mpr.refine_1.inr
False| 
inl: ∀ {a b : Prop}, a → a ∨ b
inl 
y: ?m.20498
y => 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: x = 3

h.mpr.refine_1.inr.inl
Falserw [
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: x = 3

h.mpr.refine_1.inr.inl
False
hx: ?b
hx
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: 1 = 3

h.mpr.refine_1.inr.inl
False]
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: 1 = 3

h.mpr.refine_1.inr.inl
False at 
y: ?m.20498
y
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: 1 = 3

h.mpr.refine_1.inr.inl
False;
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: 1 = 3

h.mpr.refine_1.inr.inl
False 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: x = 4

h.mpr.refine_2.inr.inl
Falsesimp at 
y: 1 = 4
y
Goals accomplished! 🐙
        
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

hz: x ∈ {3, 4}

h.mpr.refine_1.inr
False| 
inr: ∀ {a b : Prop}, b → a ∨ b
inr 
y: ?m.20267
y => 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: x ∈ {4}

h.mpr.refine_1.inr.inr
Falserw [
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: x ∈ {5}

h.mpr.refine_2.inr.inr
False
hx: ?b
hx
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: 1 ∈ {4}

h.mpr.refine_1.inr.inr
False]
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: 1 ∈ {5}

h.mpr.refine_2.inr.inr
False at 
y: ?m.20499
y
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: 1 ∈ {5}

h.mpr.refine_2.inr.inr
False;
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: 1 ∈ {5}

h.mpr.refine_2.inr.inr
False 
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

x: ℕ

hx: x ∈ {1}

y: x ∈ {4}

h.mpr.refine_1.inr.inr
Falsesimp at 
y: 1 ∈ {5}
y
Goals accomplished! 🐙

/-- ### Exercise 4.14

Show by example that for some sets `A`, `B`, and `C`, the set `A - (B - C)` is
different from `(A - B) - C`.
-/
theorem 
exercise_4_14: A \ (B \ C) ≠ (A \ B) \ C
exercise_4_14 : 
A: Set ℕ
A \ (
B: Set ℕ
B \ 
C: Set ℕ
C) ≠ (
A: Set ℕ
A \ 
B: Set ℕ
B) \ 
C: Set ℕ
C := by
Goals accomplished! 🐙
  
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

A \ (B \ C) ≠ (A \ B) \ Crw [
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

A \ (B \ C) ≠ (A \ B) \ C@
right_diff_eq_insert_one_three: ∀ {A B C : Set ℕ} {hA : A = {1, 2, 3}} {hB : B = {2, 3, 4}} {hC : C = {3, 4, 5}}, A \ (B \ C) = {1, 3}
right_diff_eq_insert_one_three 
A: Set ℕ
A 
B: Set ℕ
B 
C: Set ℕ
C 
hA: A = {1, 2, 3}
hA 
hB: B = {2, 3, 4}
hB 
hC: C = {3, 4, 5}
hC,
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

{1, 3} ≠ (A \ B) \ C
    
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

A \ (B \ C) ≠ (A \ B) \ C@
left_diff_eq_singleton_one: ∀ {A B C : Set ℕ} {hA : A = {1, 2, 3}} {hB : B = {2, 3, 4}} {hC : C = {3, 4, 5}}, (A \ B) \ C = {1}
left_diff_eq_singleton_one 
A: Set ℕ
A 
B: Set ℕ
B 
C: Set ℕ
C 
hA: A = {1, 2, 3}
hA 
hB: B = {2, 3, 4}
hB 
hC: C = {3, 4, 5}
hC
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

{1, 3} ≠ {1}
  
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

A \ (B \ C) ≠ (A \ B) \ C]
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

{1, 3} ≠ {1}
  
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

A \ (B \ C) ≠ (A \ B) \ Cintro 
h: {1, 3} = {1}
h
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

h: {1, 3} = {1}

False
  
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

A \ (B \ C) ≠ (A \ B) \ Crw [
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

h: {1, 3} = {1}

False
Set.ext_iff: ∀ {α : Type ?u.21276} {s t : Set α}, s = t ↔ ∀ (x : α), x ∈ s ↔ x ∈ t
Set.ext_iff
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

h: ∀ (x : ℕ), x ∈ {1, 3} ↔ x ∈ {1}

False]
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

h: ∀ (x : ℕ), x ∈ {1, 3} ↔ x ∈ {1}

False at 
h: {1, 3} = {1}
h
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

h: ∀ (x : ℕ), x ∈ {1, 3} ↔ x ∈ {1}

False
  
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

A \ (B \ C) ≠ (A \ B) \ Chave := 
h: ∀ (x : ℕ), x ∈ {1, 3} ↔ x ∈ {1}
h 
3: ?m.21305
3
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

h: ∀ (x : ℕ), x ∈ {1, 3} ↔ x ∈ {1}

this: 3 ∈ {1, 3} ↔ 3 ∈ {1}

False
  
A, B, C: Set ℕ

hA: A = {1, 2, 3}

hB: B = {2, 3, 4}

hC: C = {3, 4, 5}

A \ (B \ C) ≠ (A \ B) \ Csimp at 
this: ?m.21303
this
Goals accomplished! 🐙

end

/-- ### Exercise 4.16

Simplify:
`[(A ∪ B ∪ C) ∩ (A ∪ B)] - [(A ∪ (B - C)) ∩ A]`
-/
theorem 
exercise_4_16: ∀ {α : Type u_1} {A B C : Set α}, ((A ∪ B ∪ C) ∩ (A ∪ B)) \ ((A ∪ B \ C) ∩ A) = B \ A
exercise_4_16 {
A: Set α
A 
B: Set α
B 
C: Set α
C : 
Set: Type ?u.21559 → Type ?u.21559
Set 
α: ?m.21556
α}
  : ((
A: Set α
A ∪ 
B: Set α
B ∪ 
C: Set α
C) ∩ (
A: Set α
A ∪ 
B: Set α
B)) \ ((
A: Set α
A ∪ (
B: Set α
B \ 
C: Set α
C)) ∩ 
A: Set α
A) = 
B: Set α
B \ 
A: Set α
A := by
Goals accomplished! 🐙
  
α: Type u_1

A, B, C: Set α

((A ∪ B ∪ C) ∩ (A ∪ B)) \ ((A ∪ B \ C) ∩ A) = B \ Acalc ((
A: Set α
A ∪ 
B: Set α
B ∪ 
C: Set α
C) ∩ (
A: Set α
A ∪ 
B: Set α
B)) \ ((
A: Set α
A ∪ (
B: Set α
B \ 
C: Set α
C)) ∩ 
A: Set α
A)
    
_: ?m✝
_ = (
A: Set α
A ∪ 
B: Set α
B) \ ((
A: Set α
A ∪ (
B: Set α
B \ 
C: Set α
C)) ∩ 
A: Set α
A) := 
α: Type u_1

A, B, C: Set α

((A ∪ B ∪ C) ∩ (A ∪ B)) \ ((A ∪ B \ C) ∩ A) = B \ Aby
Goals accomplished! 🐙 
α: Type u_1

A, B, C: Set α

((A ∪ B ∪ C) ∩ (A ∪ B)) \ ((A ∪ B \ C) ∩ A) = (A ∪ B) \ ((A ∪ B \ C) ∩ A)rw [
α: Type u_1

A, B, C: Set α

((A ∪ B ∪ C) ∩ (A ∪ B)) \ ((A ∪ B \ C) ∩ A) = (A ∪ B) \ ((A ∪ B \ C) ∩ A)
Set.union_inter_cancel_left: ∀ {α : Type ?u.22069} {s t : Set α}, (s ∪ t) ∩ s = s
Set.union_inter_cancel_left
α: Type u_1

A, B, C: Set α

(A ∪ B) \ ((A ∪ B \ C) ∩ A) = (A ∪ B) \ ((A ∪ B \ C) ∩ A)]
Goals accomplished! 🐙
    
_: ?m✝
_ = (
A: Set α
A ∪ 
B: Set α
B) \ 
A: Set α
A := 
α: Type u_1

A, B, C: Set α

((A ∪ B ∪ C) ∩ (A ∪ B)) \ ((A ∪ B \ C) ∩ A) = B \ Aby
Goals accomplished! 🐙 
α: Type u_1

A, B, C: Set α

(A ∪ B) \ ((A ∪ B \ C) ∩ A) = (A ∪ B) \ Arw [
α: Type u_1

A, B, C: Set α

(A ∪ B) \ ((A ∪ B \ C) ∩ A) = (A ∪ B) \ A
Set.union_inter_cancel_left: ∀ {α : Type ?u.22093} {s t : Set α}, (s ∪ t) ∩ s = s
Set.union_inter_cancel_left
α: Type u_1

A, B, C: Set α

(A ∪ B) \ A = (A ∪ B) \ A]
Goals accomplished! 🐙
    
_: ?m✝
_ = 
B: Set α
B \ 
A: Set α
A := 
α: Type u_1

A, B, C: Set α

((A ∪ B ∪ C) ∩ (A ∪ B)) \ ((A ∪ B \ C) ∩ A) = B \ Aby
Goals accomplished! 🐙 
α: Type u_1

A, B, C: Set α

(A ∪ B) \ A = B \ Arw [
α: Type u_1

A, B, C: Set α

(A ∪ B) \ A = B \ A
Set.union_diff_left: ∀ {α : Type ?u.22105} {s t : Set α}, (s ∪ t) \ s = t \ s
Set.union_diff_left
α: Type u_1

A, B, C: Set α

B \ A = B \ A]
Goals accomplished! 🐙

/-! ### Exercise 4.17

Show that the following four conditions are equivalent.

(a) `A ⊆ B`
(b) `A - B = ∅`
(c) `A ∪ B = B`
(d) `A ∩ B = A`
-/

theorem 
exercise_4_17_i: ∀ {α : Type u_1} {A B : Set α}, A ⊆ B → A \ B = ∅
exercise_4_17_i {
A: Set α
A 
B: Set α
B : 
Set: Type ?u.22138 → Type ?u.22138
Set 
α: ?m.22132
α} (
h: A ⊆ B
h : 
A: Set α
A ⊆ 
B: Set α
B)
  : 
A: Set α
A \ 
B: Set α
B = 
∅: ?m.22177
∅ := by
Goals accomplished! 🐙
  
α: Type u_1

A, B: Set α

h: A ⊆ B

A \ B = ∅ext 
x: ?α
x
α: Type u_1

A, B: Set α

h: A ⊆ B

x: α

h
x ∈ A \ B ↔ x ∈ ∅
  
α: Type u_1

A, B: Set α

h: A ⊆ B

A \ B = ∅apply 
Iff.intro: ∀ {a b : Prop}, (a → b) → (b → a) → (a ↔ b)
Iff.intro
α: Type u_1

A, B: Set α

h: A ⊆ B

x: α

h.mp
x ∈ A \ B → x ∈ ∅
α: Type u_1

A, B: Set α

h: A ⊆ B

x: α

h.mpr
x ∈ ∅ → x ∈ A \ B
  
α: Type u_1

A, B: Set α

h: A ⊆ B

A \ B = ∅·
α: Type u_1

A, B: Set α

h: A ⊆ B

x: α

h.mp
x ∈ A \ B → x ∈ ∅ 
α: Type u_1

A, B: Set α

h: A ⊆ B

x: α

h.mp
x ∈ A \ B → x ∈ ∅
α: Type u_1

A, B: Set α

h: A ⊆ B

x: α

h.mpr
x ∈ ∅ → x ∈ A \ Bintro 
hx: ?a
hx
α: Type u_1

A, B: Set α

h: A ⊆ B

x: α

hx: x ∈ A \ B

h.mp
x ∈ ∅
    
α: Type u_1

A, B: Set α

h: A ⊆ B

x: α

h.mp
x ∈ A \ B → x ∈ ∅exact 
absurd: ∀ {a : Prop} {b : Sort ?u.22297}, a → ¬a → b
absurd (
h: A ⊆ B
h 
hx: ?a
hx.
left: ∀ {a b : Prop}, a ∧ b → a
left) 
hx: ?a
hx.
right: ∀ {a b : Prop}, a ∧ b → b
right
Goals accomplished! 🐙
  
α: Type u_1

A, B: Set α

h: A ⊆ B

A \ B = ∅·
α: Type u_1

A, B: Set α

h: A ⊆ B

x: α

h.mpr
x ∈ ∅ → x ∈ A \ B 
α: Type u_1

A, B: Set α

h: A ⊆ B

x: α

h.mpr
x ∈ ∅ → x ∈ A \ Bintro 
hx: ?b
hx
α: Type u_1

A, B: Set α

h: A ⊆ B

x: α

hx: x ∈ ∅

h.mpr
x ∈ A \ B
    
α: Type u_1

A, B: Set α

h: A ⊆ B

x: α

h.mpr
x ∈ ∅ → x ∈ A \ Bexact 
False.elim: ∀ {C : Sort ?u.22317}, False → C
False.elim 
hx: ?b
hx
Goals accomplished! 🐙

theorem 
exercise_4_17_ii: ∀ {α : Type u_1} {A B : Set α}, A \ B = ∅ → A ∪ B = B
exercise_4_17_ii {
A: Set α
A 
B: Set α
B : 
Set: Type ?u.22335 → Type ?u.22335
Set 
α: ?m.22329
α} (
h: A \ B = ∅
h : 
A: Set α
A \ 
B: Set α
B = 
∅: ?m.22357
∅)
  : 
A: Set α
A ∪ 
B: Set α
B = 
B: Set α
B := by
Goals accomplished! 🐙
  
α: Type u_1

A, B: Set α

h: A \ B = ∅

A ∪ B = Bsuffices 
A: Set α
A ⊆ 
B: Set α
B from 
Set.left_subset_union_eq_self: ∀ {α : Type ?u.22474} {A B : Set α}, A ⊆ B → A ∪ B = B
Set.left_subset_union_eq_self 
this: A ⊆ B
this
α: Type u_1

A, B: Set α

h: A \ B = ∅

A ⊆ B
  
α: Type u_1

A, B: Set α

h: A \ B = ∅

A ∪ B = Bshow ∀ 
t: ?m.22491
t, 
t: ?m.22491
t ∈ 
A: Set α
A → 
t: ?m.22491
t ∈ 
B: Set α
B
α: Type u_1

A, B: Set α

h: A \ B = ∅

∀ (t : α), t ∈ A → t ∈ B
  
α: Type u_1

A, B: Set α

h: A \ B = ∅

A ∪ B = Bintro 
t: α
t 
ht: t ∈ A
ht
α: Type u_1

A, B: Set α

h: A \ B = ∅

t: α

ht: t ∈ A

t ∈ B
  
α: Type u_1

A, B: Set α

h: A \ B = ∅

A ∪ B = Brw [
α: Type u_1

A, B: Set α

h: A \ B = ∅

t: α

ht: t ∈ A

t ∈ B
Set.ext_iff: ∀ {α : Type ?u.22563} {s t : Set α}, s = t ↔ ∀ (x : α), x ∈ s ↔ x ∈ t
Set.ext_iff
α: Type u_1

A, B: Set α

h: ∀ (x : α), x ∈ A \ B ↔ x ∈ ∅

t: α

ht: t ∈ A

t ∈ B]
α: Type u_1

A, B: Set α

h: ∀ (x : α), x ∈ A \ B ↔ x ∈ ∅

t: α

ht: t ∈ A

t ∈ B at 
h: A \ B = ∅
h
α: Type u_1

A, B: Set α

h: ∀ (x : α), x ∈ A \ B ↔ x ∈ ∅

t: α

ht: t ∈ A

t ∈ B
  
α: Type u_1

A, B: Set α

h: A \ B = ∅

A ∪ B = Bby_contra 
nt: ?m.22728
nt
α: Type u_1

A, B: Set α

h: ∀ (x : α), x ∈ A \ B ↔ x ∈ ∅

t: α

ht: t ∈ A

nt: ¬t ∈ B

False
  
α: Type u_1

A, B: Set α

h: A \ B = ∅

A ∪ B = Bexact (
h: ∀ (x : α), x ∈ A \ B ↔ x ∈ ∅
h 
t: α
t).
mp: ∀ {a b : Prop}, (a ↔ b) → a → b
mp ⟨
ht: t ∈ A
ht, 
nt: ?m.22728
nt⟩
Goals accomplished! 🐙

theorem 
exercise_4_17_iii: ∀ {α : Type u_1} {A B : Set α}, A ∪ B = B → A ∩ B = A
exercise_4_17_iii {
A: Set α
A 
B: Set α
B : 
Set: Type ?u.22771 → Type ?u.22771
Set 
α: ?m.22765
α} (
h: A ∪ B = B
h : 
A: Set α
A ∪ 
B: Set α
B = 
B: Set α
B)
  : 
A: Set α
A ∩ 
B: Set α
B = 
A: Set α
A := by
Goals accomplished! 🐙
  
α: Type u_1

A, B: Set α

h: A ∪ B = B

A ∩ B = Asuffices 
A: Set α
A ⊆ 
B: Set α
B from 
Set.inter_eq_left_iff_subset: ∀ {α : Type ?u.22832} {s t : Set α}, s ∩ t = s ↔ s ⊆ t
Set.inter_eq_left_iff_subset.
mpr: ∀ {a b : Prop}, (a ↔ b) → b → a
mpr 
this: A ⊆ B
this
α: Type u_1

A, B: Set α

h: A ∪ B = B

A ⊆ B
  
α: Type u_1

A, B: Set α

h: A ∪ B = B

A ∩ B = Aexact 
Set.union_eq_right_iff_subset: ∀ {α : Type ?u.22851} {s t : Set α}, s ∪ t = t ↔ s ⊆ t
Set.union_eq_right_iff_subset.
mp: ∀ {a b : Prop}, (a ↔ b) → a → b
mp 
h: A ∪ B = B
h
Goals accomplished! 🐙

theorem 
exercise_4_17_iv: ∀ {α : Type u_1} {A B : Set α}, A ∩ B = A → A ⊆ B
exercise_4_17_iv {
A: Set α
A 
B: Set α
B : 
Set: Type ?u.22879 → Type ?u.22879
Set 
α: ?m.22873
α} (
h: A ∩ B = A
h : 
A: Set α
A ∩ 
B: Set α
B = 
A: Set α
A)
  : 
A: Set α
A ⊆ 
B: Set α
B := 
Set.inter_eq_left_iff_subset: ∀ {α : Type ?u.22924} {s t : Set α}, s ∩ t = s ↔ s ⊆ t
Set.inter_eq_left_iff_subset.
mp: ∀ {a b : Prop}, (a ↔ b) → a → b
mp 
h: A ∩ B = A
h

/-- ### Exercise 4.19

Is `𝒫 (A - B)` always equal to `𝒫 A - 𝒫 B`? Is it ever equal to `𝒫 A - 𝒫 B`?
-/
theorem 
exercise_4_19: ∀ {α : Type u_1} {A B : Set α}, 𝒫(A \ B) ≠ 𝒫 A \ 𝒫 B
exercise_4_19 {
A: Set α
A 
B: Set α
B : 
Set: Type ?u.22950 → Type ?u.22950
Set 
α: ?m.22947
α}
  : 𝒫 (
A: Set α
A \ 
B: Set α
B) ≠ (𝒫 
A: Set α
A) \ (𝒫 
B: Set α
B) := by
Goals accomplished! 🐙
  
α: Type u_1

A, B: Set α

𝒫(A \ B) ≠ 𝒫 A \ 𝒫 Bintro 
h: 𝒫(A \ B) = 𝒫 A \ 𝒫 B
h
α: Type u_1

A, B: Set α

h: 𝒫(A \ B) = 𝒫 A \ 𝒫 B

False
  
α: Type u_1

A, B: Set α

𝒫(A \ B) ≠ 𝒫 A \ 𝒫 Bhave 
he: ∅ ∈ 𝒫(A \ B)
he : 
∅: ?m.23054
∅ ∈ 𝒫 (
A: Set α
A \ 
B: Set α
B) := 
α: Type u_1

A, B: Set α

h: 𝒫(A \ B) = 𝒫 A \ 𝒫 B

Falseby
Goals accomplished! 🐙 
α: Type u_1

A, B: Set α

h: 𝒫(A \ B) = 𝒫 A \ 𝒫 B

∅ ∈ 𝒫(A \ B)simp
Goals accomplished! 🐙
  
α: Type u_1

A, B: Set α

𝒫(A \ B) ≠ 𝒫 A \ 𝒫 Bhave 
ne: ¬∅ ∈ 𝒫 A \ 𝒫 B
ne : 
∅: ?m.23311
∅ ∉ (𝒫 
A: Set α
A) \ (𝒫 
B: Set α
B) := 
α: Type u_1

A, B: Set α

h: 𝒫(A \ B) = 𝒫 A \ 𝒫 B

he: ∅ ∈ 𝒫(A \ B)

Falseby
Goals accomplished! 🐙 
α: Type u_1

A, B: Set α

h: 𝒫(A \ B) = 𝒫 A \ 𝒫 B

he: ∅ ∈ 𝒫(A \ B)

¬∅ ∈ 𝒫 A \ 𝒫 Bsimp
Goals accomplished! 🐙
  
α: Type u_1

A, B: Set α

𝒫(A \ B) ≠ 𝒫 A \ 𝒫 Brw [
α: Type u_1

A, B: Set α

h: 𝒫(A \ B) = 𝒫 A \ 𝒫 B

he: ∅ ∈ 𝒫(A \ B)

ne: ¬∅ ∈ 𝒫 A \ 𝒫 B

False
Set.ext_iff: ∀ {α : Type ?u.23523} {s t : Set α}, s = t ↔ ∀ (x : α), x ∈ s ↔ x ∈ t
Set.ext_iff
α: Type u_1

A, B: Set α

h: ∀ (x : Set α), x ∈ 𝒫(A \ B) ↔ x ∈ 𝒫 A \ 𝒫 B

he: ∅ ∈ 𝒫(A \ B)

ne: ¬∅ ∈ 𝒫 A \ 𝒫 B

False]
α: Type u_1

A, B: Set α

h: ∀ (x : Set α), x ∈ 𝒫(A \ B) ↔ x ∈ 𝒫 A \ 𝒫 B

he: ∅ ∈ 𝒫(A \ B)

ne: ¬∅ ∈ 𝒫 A \ 𝒫 B

False at 
h: 𝒫(A \ B) = 𝒫 A \ 𝒫 B
h
α: Type u_1

A, B: Set α

h: ∀ (x : Set α), x ∈ 𝒫(A \ B) ↔ x ∈ 𝒫 A \ 𝒫 B

he: ∅ ∈ 𝒫(A \ B)

ne: ¬∅ ∈ 𝒫 A \ 𝒫 B

False
  
α: Type u_1

A, B: Set α

𝒫(A \ B) ≠ 𝒫 A \ 𝒫 Bhave := 
h: ∀ (x : Set α), x ∈ 𝒫(A \ B) ↔ x ∈ 𝒫 A \ 𝒫 B
h 
∅: ?m.23557
∅
α: Type u_1

A, B: Set α

h: ∀ (x : Set α), x ∈ 𝒫(A \ B) ↔ x ∈ 𝒫 A \ 𝒫 B

he: ∅ ∈ 𝒫(A \ B)

ne: ¬∅ ∈ 𝒫 A \ 𝒫 B

this: ∅ ∈ 𝒫(A \ B) ↔ ∅ ∈ 𝒫 A \ 𝒫 B

False
  
α: Type u_1

A, B: Set α

𝒫(A \ B) ≠ 𝒫 A \ 𝒫 Bexact 
absurd: ∀ {a : Prop} {b : Sort ?u.23599}, a → ¬a → b
absurd (
this: ?m.23555
this.
mp: ∀ {a b : Prop}, (a ↔ b) → a → b
mp 
he: ∅ ∈ 𝒫(A \ B)
he) 
ne: ¬∅ ∈ 𝒫 A \ 𝒫 B
ne
Goals accomplished! 🐙

/-- ### Exercise 4.20

Let `A`, `B`, and `C` be sets such that `A ∪ B = A ∪ C` and `A ∩ B = A ∩ C`.
Show that `B = C`.
-/
theorem 
exercise_4_20: ∀ {α : Type u_1} {A B C : Set α}, A ∪ B = A ∪ C → A ∩ B = A ∩ C → B = C
exercise_4_20 {
A: Set α
A 
B: Set α
B 
C: Set α
C : 
Set: Type ?u.23625 → Type ?u.23625
Set 
α: ?m.23619
α}
  (
hu: A ∪ B = A ∪ C
hu : 
A: Set α
A ∪ 
B: Set α
B = 
A: Set α
A ∪ 
C: Set α
C) (
hi: A ∩ B = A ∩ C
hi : 
A: Set α
A ∩ 
B: Set α
B = 
A: Set α
A ∩ 
C: Set α
C) : 
B: Set α
B = 
C: Set α
C := by
Goals accomplished! 🐙
  
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

B = Cext 
x: ?α
x
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

h
x ∈ B ↔ x ∈ C
  
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

B = Capply 
Iff.intro: ∀ {a b : Prop}, (a → b) → (b → a) → (a ↔ b)
Iff.intro
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

h.mp
x ∈ B → x ∈ C
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

h.mpr
x ∈ C → x ∈ B
  
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

B = C·
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

h.mp
x ∈ B → x ∈ C 
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

h.mp
x ∈ B → x ∈ C
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

h.mpr
x ∈ C → x ∈ Bintro 
hB: ?a
hB
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

h.mp
x ∈ C
    
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

h.mp
x ∈ B → x ∈ Cby_cases 
hA: ?m.23757
hA : 
x: ?α
x ∈ 
A: Set α
A
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: x ∈ A

pos
x ∈ C
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: ¬x ∈ A

neg
x ∈ C
    
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

h.mp
x ∈ B → x ∈ C·
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: x ∈ A

pos
x ∈ C 
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: x ∈ A

pos
x ∈ C
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: ¬x ∈ A

neg
x ∈ Chave : 
x: ?α
x ∈ 
A: Set α
A ∩ 
B: Set α
B := 
Set.mem_inter: ∀ {α : Type ?u.23805} {x : α} {a b : Set α}, x ∈ a → x ∈ b → x ∈ a ∩ b
Set.mem_inter 
hA: ?m.23750
hA 
hB: ?a
hB
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: x ∈ A

this: x ∈ A ∩ B

pos
x ∈ C
      
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: x ∈ A

pos
x ∈ Crw [
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: x ∈ A

this: x ∈ A ∩ B

pos
x ∈ C
hi: A ∩ B = A ∩ C
hi
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: x ∈ A

this: x ∈ A ∩ C

pos
x ∈ C]
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: x ∈ A

this: x ∈ A ∩ C

pos
x ∈ C at 
this: x ∈ A ∩ B
this
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: x ∈ A

this: x ∈ A ∩ C

pos
x ∈ C
      
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: x ∈ A

pos
x ∈ Cexact 
this: x ∈ A ∩ C
this.
right: ∀ {a b : Prop}, a ∧ b → b
right
Goals accomplished! 🐙
    
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

h.mp
x ∈ B → x ∈ C·
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: ¬x ∈ A

neg
x ∈ C 
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: ¬x ∈ A

neg
x ∈ Chave : 
x: ?α
x ∈ 
A: Set α
A ∪ 
B: Set α
B := 
Set.mem_union_right: ∀ {α : Type ?u.23878} {x : α} {b : Set α} (a : Set α), x ∈ b → x ∈ a ∪ b
Set.mem_union_right 
A: Set α
A 
hB: ?a
hB
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: ¬x ∈ A

this: x ∈ A ∪ B

neg
x ∈ C
      
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: ¬x ∈ A

neg
x ∈ Crw [
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: ¬x ∈ A

this: x ∈ A ∪ B

neg
x ∈ C
hu: A ∪ B = A ∪ C
hu
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: ¬x ∈ A

this: x ∈ A ∪ C

neg
x ∈ C]
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: ¬x ∈ A

this: x ∈ A ∪ C

neg
x ∈ C at 
this: x ∈ A ∪ B
this
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: ¬x ∈ A

this: x ∈ A ∪ C

neg
x ∈ C
      
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: ¬x ∈ A

neg
x ∈ Cexact 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
this: x ∈ A ∪ C
this (
absurd: ∀ {a : Prop} {b : Sort ?u.23917}, a → ¬a → b
absurd · 
hA: ?m.23757
hA) (
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: ¬x ∈ A

this: x ∈ A ∪ C

neg
x ∈ Cby
Goals accomplished! 🐙 
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hB: x ∈ B

hA: ¬x ∈ A

this: x ∈ A ∪ C

x ∈ C → x ∈ Csimp
Goals accomplished! 🐙)
Goals accomplished! 🐙
  
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

B = C·
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

h.mpr
x ∈ C → x ∈ B 
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

h.mpr
x ∈ C → x ∈ Bintro 
hC: ?b
hC
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

h.mpr
x ∈ B
    
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

h.mpr
x ∈ C → x ∈ Bby_cases 
hA: ?m.24111
hA : 
x: ?α
x ∈ 
A: Set α
A
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: x ∈ A

pos
x ∈ B
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: ¬x ∈ A

neg
x ∈ B
    
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

h.mpr
x ∈ C → x ∈ B·
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: x ∈ A

pos
x ∈ B 
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: x ∈ A

pos
x ∈ B
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: ¬x ∈ A

neg
x ∈ Bhave : 
x: ?α
x ∈ 
A: Set α
A ∩ 
C: Set α
C := 
Set.mem_inter: ∀ {α : Type ?u.24159} {x : α} {a b : Set α}, x ∈ a → x ∈ b → x ∈ a ∩ b
Set.mem_inter 
hA: ?m.24104
hA 
hC: ?b
hC
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: x ∈ A

this: x ∈ A ∩ C

pos
x ∈ B
      
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: x ∈ A

pos
x ∈ Brw [
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: x ∈ A

this: x ∈ A ∩ C

pos
x ∈ B← 
hi: A ∩ B = A ∩ C
hi
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: x ∈ A

this: x ∈ A ∩ B

pos
x ∈ B]
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: x ∈ A

this: x ∈ A ∩ B

pos
x ∈ B at 
this: x ∈ A ∩ C
this
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: x ∈ A

this: x ∈ A ∩ B

pos
x ∈ B
      
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: x ∈ A

pos
x ∈ Bexact 
this: x ∈ A ∩ B
this.
right: ∀ {a b : Prop}, a ∧ b → b
right
Goals accomplished! 🐙
    
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

h.mpr
x ∈ C → x ∈ B·
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: ¬x ∈ A

neg
x ∈ B 
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: ¬x ∈ A

neg
x ∈ Bhave : 
x: ?α
x ∈ 
A: Set α
A ∪ 
C: Set α
C := 
Set.mem_union_right: ∀ {α : Type ?u.24232} {x : α} {b : Set α} (a : Set α), x ∈ b → x ∈ a ∪ b
Set.mem_union_right 
A: Set α
A 
hC: ?b
hC
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: ¬x ∈ A

this: x ∈ A ∪ C

neg
x ∈ B
      
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: ¬x ∈ A

neg
x ∈ Brw [
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: ¬x ∈ A

this: x ∈ A ∪ C

neg
x ∈ B← 
hu: A ∪ B = A ∪ C
hu
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: ¬x ∈ A

this: x ∈ A ∪ B

neg
x ∈ B]
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: ¬x ∈ A

this: x ∈ A ∪ B

neg
x ∈ B at 
this: x ∈ A ∪ C
this
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: ¬x ∈ A

this: x ∈ A ∪ B

neg
x ∈ B
      
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: ¬x ∈ A

neg
x ∈ Bexact 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
this: x ∈ A ∪ B
this (
absurd: ∀ {a : Prop} {b : Sort ?u.24271}, a → ¬a → b
absurd · 
hA: ?m.24111
hA) (
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: ¬x ∈ A

this: x ∈ A ∪ B

neg
x ∈ Bby
Goals accomplished! 🐙 
α: Type u_1

A, B, C: Set α

hu: A ∪ B = A ∪ C

hi: A ∩ B = A ∩ C

x: α

hC: x ∈ C

hA: ¬x ∈ A

this: x ∈ A ∪ B

x ∈ B → x ∈ Bsimp
Goals accomplished! 🐙)
Goals accomplished! 🐙

/-- ### Exercise 4.21

Show that `⋃ (A ∪ B) = (⋃ A) ∪ (⋃ B)`.
-/
theorem 
exercise_4_21: ∀ {α : Type u_1} {A B : Set (Set α)}, ⋃₀ (A ∪ B) = ⋃₀ A ∪ ⋃₀ B
exercise_4_21 {
A: Set (Set α)
A 
B: Set (Set α)
B : 
Set: Type ?u.24463 → Type ?u.24463
Set (
Set: Type ?u.24468 → Type ?u.24468
Set 
α: ?m.24460
α)}
  : ⋃₀ (
A: Set (Set α)
A ∪ 
B: Set (Set α)
B) = (⋃₀ 
A: Set (Set α)
A) ∪ (⋃₀ 
B: Set (Set α)
B) := by
Goals accomplished! 🐙
  
α: Type u_1

A, B: Set (Set α)

⋃₀ (A ∪ B) = ⋃₀ A ∪ ⋃₀ Bext 
x: ?α
x
α: Type u_1

A, B: Set (Set α)

x: α

h
x ∈ ⋃₀ (A ∪ B) ↔ x ∈ ⋃₀ A ∪ ⋃₀ B
  
α: Type u_1

A, B: Set (Set α)

⋃₀ (A ∪ B) = ⋃₀ A ∪ ⋃₀ Bapply 
Iff.intro: ∀ {a b : Prop}, (a → b) → (b → a) → (a ↔ b)
Iff.intro
α: Type u_1

A, B: Set (Set α)

x: α

h.mp
x ∈ ⋃₀ (A ∪ B) → x ∈ ⋃₀ A ∪ ⋃₀ B
α: Type u_1

A, B: Set (Set α)

x: α

h.mpr
x ∈ ⋃₀ A ∪ ⋃₀ B → x ∈ ⋃₀ (A ∪ B)
  
α: Type u_1

A, B: Set (Set α)

⋃₀ (A ∪ B) = ⋃₀ A ∪ ⋃₀ B·
α: Type u_1

A, B: Set (Set α)

x: α

h.mp
x ∈ ⋃₀ (A ∪ B) → x ∈ ⋃₀ A ∪ ⋃₀ B 
α: Type u_1

A, B: Set (Set α)

x: α

h.mp
x ∈ ⋃₀ (A ∪ B) → x ∈ ⋃₀ A ∪ ⋃₀ B
α: Type u_1

A, B: Set (Set α)

x: α

h.mpr
x ∈ ⋃₀ A ∪ ⋃₀ B → x ∈ ⋃₀ (A ∪ B)intro 
hx: ?a
hx
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ (A ∪ B)

h.mp
x ∈ ⋃₀ A ∪ ⋃₀ B
    
α: Type u_1

A, B: Set (Set α)

x: α

h.mp
x ∈ ⋃₀ (A ∪ B) → x ∈ ⋃₀ A ∪ ⋃₀ Bhave ⟨
t: Set α
t, 
ht: t ∈ A ∪ B ∧ x ∈ t
ht⟩ : ∃ 
t: ?m.24561
t, 
t: ?m.24561
t ∈ 
A: Set (Set α)
A ∪ 
B: Set (Set α)
B ∧ 
x: ?α
x ∈ 
t: ?m.24561
t := 
hx: ?a
hx
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ (A ∪ B)

t: Set α

ht: t ∈ A ∪ B ∧ x ∈ t

h.mp
x ∈ ⋃₀ A ∪ ⋃₀ B
    
α: Type u_1

A, B: Set (Set α)

x: α

h.mp
x ∈ ⋃₀ (A ∪ B) → x ∈ ⋃₀ A ∪ ⋃₀ Bapply 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
ht: t ∈ A ∪ B ∧ x ∈ t
ht.
left: ∀ {a b : Prop}, a ∧ b → a
left
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ (A ∪ B)

t: Set α

ht: t ∈ A ∪ B ∧ x ∈ t

h.mp.left
t ∈ A → x ∈ ⋃₀ A ∪ ⋃₀ B
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ (A ∪ B)

t: Set α

ht: t ∈ A ∪ B ∧ x ∈ t

h.mp.right
t ∈ B → x ∈ ⋃₀ A ∪ ⋃₀ B
    
α: Type u_1

A, B: Set (Set α)

x: α

h.mp
x ∈ ⋃₀ (A ∪ B) → x ∈ ⋃₀ A ∪ ⋃₀ B·
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ (A ∪ B)

t: Set α

ht: t ∈ A ∪ B ∧ x ∈ t

h.mp.left
t ∈ A → x ∈ ⋃₀ A ∪ ⋃₀ B 
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ (A ∪ B)

t: Set α

ht: t ∈ A ∪ B ∧ x ∈ t

h.mp.left
t ∈ A → x ∈ ⋃₀ A ∪ ⋃₀ B
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ (A ∪ B)

t: Set α

ht: t ∈ A ∪ B ∧ x ∈ t

h.mp.right
t ∈ B → x ∈ ⋃₀ A ∪ ⋃₀ Bintro 
hA: t ∈ A
hA
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ (A ∪ B)

t: Set α

ht: t ∈ A ∪ B ∧ x ∈ t

hA: t ∈ A

h.mp.left
x ∈ ⋃₀ A ∪ ⋃₀ B
      
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ (A ∪ B)

t: Set α

ht: t ∈ A ∪ B ∧ x ∈ t

h.mp.left
t ∈ A → x ∈ ⋃₀ A ∪ ⋃₀ Bexact 
Or.inl: ∀ {a b : Prop}, a → a ∨ b
Or.inl ⟨
t: Set α
t, ⟨
hA: t ∈ A
hA, 
ht: t ∈ A ∪ B ∧ x ∈ t
ht.
right: ∀ {a b : Prop}, a ∧ b → b
right⟩⟩
Goals accomplished! 🐙
    
α: Type u_1

A, B: Set (Set α)

x: α

h.mp
x ∈ ⋃₀ (A ∪ B) → x ∈ ⋃₀ A ∪ ⋃₀ B·
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ (A ∪ B)

t: Set α

ht: t ∈ A ∪ B ∧ x ∈ t

h.mp.right
t ∈ B → x ∈ ⋃₀ A ∪ ⋃₀ B 
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ (A ∪ B)

t: Set α

ht: t ∈ A ∪ B ∧ x ∈ t

h.mp.right
t ∈ B → x ∈ ⋃₀ A ∪ ⋃₀ Bintro 
hB: t ∈ B
hB
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ (A ∪ B)

t: Set α

ht: t ∈ A ∪ B ∧ x ∈ t

hB: t ∈ B

h.mp.right
x ∈ ⋃₀ A ∪ ⋃₀ B
      
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ (A ∪ B)

t: Set α

ht: t ∈ A ∪ B ∧ x ∈ t

h.mp.right
t ∈ B → x ∈ ⋃₀ A ∪ ⋃₀ Bexact 
Or.inr: ∀ {a b : Prop}, b → a ∨ b
Or.inr ⟨
t: Set α
t, ⟨
hB: t ∈ B
hB, 
ht: t ∈ A ∪ B ∧ x ∈ t
ht.
right: ∀ {a b : Prop}, a ∧ b → b
right⟩⟩
Goals accomplished! 🐙
  
α: Type u_1

A, B: Set (Set α)

⋃₀ (A ∪ B) = ⋃₀ A ∪ ⋃₀ B·
α: Type u_1

A, B: Set (Set α)

x: α

h.mpr
x ∈ ⋃₀ A ∪ ⋃₀ B → x ∈ ⋃₀ (A ∪ B) 
α: Type u_1

A, B: Set (Set α)

x: α

h.mpr
x ∈ ⋃₀ A ∪ ⋃₀ B → x ∈ ⋃₀ (A ∪ B)intro 
hx: ?b
hx
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ A ∪ ⋃₀ B

h.mpr
x ∈ ⋃₀ (A ∪ B)
    
α: Type u_1

A, B: Set (Set α)

x: α

h.mpr
x ∈ ⋃₀ A ∪ ⋃₀ B → x ∈ ⋃₀ (A ∪ B)apply 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
hx: ?b
hx
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ A ∪ ⋃₀ B

h.mpr.left
x ∈ ⋃₀ A → x ∈ ⋃₀ (A ∪ B)
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ A ∪ ⋃₀ B

h.mpr.right
x ∈ ⋃₀ B → x ∈ ⋃₀ (A ∪ B)
    
α: Type u_1

A, B: Set (Set α)

x: α

h.mpr
x ∈ ⋃₀ A ∪ ⋃₀ B → x ∈ ⋃₀ (A ∪ B)·
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ A ∪ ⋃₀ B

h.mpr.left
x ∈ ⋃₀ A → x ∈ ⋃₀ (A ∪ B) 
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ A ∪ ⋃₀ B

h.mpr.left
x ∈ ⋃₀ A → x ∈ ⋃₀ (A ∪ B)
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ A ∪ ⋃₀ B

h.mpr.right
x ∈ ⋃₀ B → x ∈ ⋃₀ (A ∪ B)intro 
hA: x ∈ ⋃₀ A
hA
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ A ∪ ⋃₀ B

hA: x ∈ ⋃₀ A

h.mpr.left
x ∈ ⋃₀ (A ∪ B)
      
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ A ∪ ⋃₀ B

h.mpr.left
x ∈ ⋃₀ A → x ∈ ⋃₀ (A ∪ B)have ⟨
t: Set α
t, 
ht: t ∈ A ∧ x ∈ t
ht⟩ : ∃ 
t: ?m.24788
t, 
t: ?m.24788
t ∈ 
A: Set (Set α)
A ∧ 
x: ?α
x ∈ 
t: ?m.24788
t := 
hA: x ∈ ⋃₀ A
hA
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ A ∪ ⋃₀ B

hA: x ∈ ⋃₀ A

t: Set α

ht: t ∈ A ∧ x ∈ t

h.mpr.left
x ∈ ⋃₀ (A ∪ B)
      
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ A ∪ ⋃₀ B

h.mpr.left
x ∈ ⋃₀ A → x ∈ ⋃₀ (A ∪ B)exact ⟨
t: Set α
t, ⟨
Set.mem_union_left: ∀ {α : Type ?u.24932} {x : α} {a : Set α} (b : Set α), x ∈ a → x ∈ a ∪ b
Set.mem_union_left 
B: Set (Set α)
B 
ht: t ∈ A ∧ x ∈ t
ht.
left: ∀ {a b : Prop}, a ∧ b → a
left, 
ht: t ∈ A ∧ x ∈ t
ht.
right: ∀ {a b : Prop}, a ∧ b → b
right⟩⟩
Goals accomplished! 🐙
    
α: Type u_1

A, B: Set (Set α)

x: α

h.mpr
x ∈ ⋃₀ A ∪ ⋃₀ B → x ∈ ⋃₀ (A ∪ B)·
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ A ∪ ⋃₀ B

h.mpr.right
x ∈ ⋃₀ B → x ∈ ⋃₀ (A ∪ B) 
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ A ∪ ⋃₀ B

h.mpr.right
x ∈ ⋃₀ B → x ∈ ⋃₀ (A ∪ B)intro 
hB: x ∈ ⋃₀ B
hB
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ A ∪ ⋃₀ B

hB: x ∈ ⋃₀ B

h.mpr.right
x ∈ ⋃₀ (A ∪ B)
      
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ A ∪ ⋃₀ B

h.mpr.right
x ∈ ⋃₀ B → x ∈ ⋃₀ (A ∪ B)have ⟨
t: Set α
t, 
ht: t ∈ B ∧ x ∈ t
ht⟩ : ∃ 
t: ?m.24959
t, 
t: ?m.24959
t ∈ 
B: Set (Set α)
B ∧ 
x: ?α
x ∈ 
t: ?m.24959
t := 
hB: x ∈ ⋃₀ B
hB
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ A ∪ ⋃₀ B

hB: x ∈ ⋃₀ B

t: Set α

ht: t ∈ B ∧ x ∈ t

h.mpr.right
x ∈ ⋃₀ (A ∪ B)
      
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋃₀ A ∪ ⋃₀ B

h.mpr.right
x ∈ ⋃₀ B → x ∈ ⋃₀ (A ∪ B)exact ⟨
t: Set α
t, ⟨
Set.mem_union_right: ∀ {α : Type ?u.25089} {x : α} {b : Set α} (a : Set α), x ∈ b → x ∈ a ∪ b
Set.mem_union_right 
A: Set (Set α)
A 
ht: t ∈ B ∧ x ∈ t
ht.
left: ∀ {a b : Prop}, a ∧ b → a
left, 
ht: t ∈ B ∧ x ∈ t
ht.
right: ∀ {a b : Prop}, a ∧ b → b
right⟩⟩
Goals accomplished! 🐙

/-- ### Exercise 4.22

Show that if `A` and `B` are nonempty sets, then `⋂ (A ∪ B) = ⋂ A ∩ ⋂ B`.
-/
theorem 
exercise_4_22: ∀ {α : Type u_1} {A B : Set (Set α)}, ⋂₀ (A ∪ B) = ⋂₀ A ∩ ⋂₀ B
exercise_4_22 {
A: Set (Set α)
A 
B: Set (Set α)
B : 
Set: Type ?u.25132 → Type ?u.25132
Set (
Set: Type ?u.25129 → Type ?u.25129
Set 
α: ?m.25125
α)}
  : ⋂₀ (
A: Set (Set α)
A ∪ 
B: Set (Set α)
B) = ⋂₀ 
A: Set (Set α)
A ∩ ⋂₀ 
B: Set (Set α)
B := by
Goals accomplished! 🐙
  
α: Type u_1

A, B: Set (Set α)

⋂₀ (A ∪ B) = ⋂₀ A ∩ ⋂₀ Bext 
x: ?α
x
α: Type u_1

A, B: Set (Set α)

x: α

h
x ∈ ⋂₀ (A ∪ B) ↔ x ∈ ⋂₀ A ∩ ⋂₀ B
  
α: Type u_1

A, B: Set (Set α)

⋂₀ (A ∪ B) = ⋂₀ A ∩ ⋂₀ Bapply 
Iff.intro: ∀ {a b : Prop}, (a → b) → (b → a) → (a ↔ b)
Iff.intro
α: Type u_1

A, B: Set (Set α)

x: α

h.mp
x ∈ ⋂₀ (A ∪ B) → x ∈ ⋂₀ A ∩ ⋂₀ B
α: Type u_1

A, B: Set (Set α)

x: α

h.mpr
x ∈ ⋂₀ A ∩ ⋂₀ B → x ∈ ⋂₀ (A ∪ B)
  
α: Type u_1

A, B: Set (Set α)

⋂₀ (A ∪ B) = ⋂₀ A ∩ ⋂₀ B·
α: Type u_1

A, B: Set (Set α)

x: α

h.mp
x ∈ ⋂₀ (A ∪ B) → x ∈ ⋂₀ A ∩ ⋂₀ B 
α: Type u_1

A, B: Set (Set α)

x: α

h.mp
x ∈ ⋂₀ (A ∪ B) → x ∈ ⋂₀ A ∩ ⋂₀ B
α: Type u_1

A, B: Set (Set α)

x: α

h.mpr
x ∈ ⋂₀ A ∩ ⋂₀ B → x ∈ ⋂₀ (A ∪ B)intro 
hx: ?a
hx
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ (A ∪ B)

h.mp
x ∈ ⋂₀ A ∩ ⋂₀ B
    
α: Type u_1

A, B: Set (Set α)

x: α

h.mp
x ∈ ⋂₀ (A ∪ B) → x ∈ ⋂₀ A ∩ ⋂₀ Bhave : ∀ 
t: Set α
t : 
Set: Type ?u.25226 → Type ?u.25226
Set 
α: Type u_1
α, 
t: Set α
t ∈ 
A: Set (Set α)
A ∪ 
B: Set (Set α)
B → 
x: ?α
x ∈ 
t: Set α
t := 
hx: ?a
hx
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ (A ∪ B)

this: ∀ (t : Set α), t ∈ A ∪ B → x ∈ t

h.mp
x ∈ ⋂₀ A ∩ ⋂₀ B
    
α: Type u_1

A, B: Set (Set α)

x: α

h.mp
x ∈ ⋂₀ (A ∪ B) → x ∈ ⋂₀ A ∩ ⋂₀ Bshow (∀ 
t: Set α
t : 
Set: Type ?u.25308 → Type ?u.25308
Set 
α: Type u_1
α, 
t: Set α
t ∈ 
A: Set (Set α)
A → 
x: ?α
x ∈ 
t: Set α
t) ∧ (∀ 
t: Set α
t : 
Set: Type ?u.25349 → Type ?u.25349
Set 
α: Type u_1
α, 
t: Set α
t ∈ 
B: Set (Set α)
B → 
x: ?α
x ∈ 
t: Set α
t)
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ (A ∪ B)

this: ∀ (t : Set α), t ∈ A ∪ B → x ∈ t

h.mp
(∀ (t : Set α), t ∈ A → x ∈ t) ∧ ∀ (t : Set α), t ∈ B → x ∈ t
    
α: Type u_1

A, B: Set (Set α)

x: α

h.mp
x ∈ ⋂₀ (A ∪ B) → x ∈ ⋂₀ A ∩ ⋂₀ Brw [
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ (A ∪ B)

this: ∀ (t : Set α), t ∈ A ∪ B → x ∈ t

h.mp
(∀ (t : Set α), t ∈ A → x ∈ t) ∧ ∀ (t : Set α), t ∈ B → x ∈ t← 
forall_and: ∀ {α : Sort ?u.25402} {p q : α → Prop}, (∀ (x : α), p x ∧ q x) ↔ (∀ (x : α), p x) ∧ ∀ (x : α), q x
forall_and
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ (A ∪ B)

this: ∀ (t : Set α), t ∈ A ∪ B → x ∈ t

h.mp
∀ (x_1 : Set α), (x_1 ∈ A → x ∈ x_1) ∧ (x_1 ∈ B → x ∈ x_1)]
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ (A ∪ B)

this: ∀ (t : Set α), t ∈ A ∪ B → x ∈ t

h.mp
∀ (x_1 : Set α), (x_1 ∈ A → x ∈ x_1) ∧ (x_1 ∈ B → x ∈ x_1)
    
α: Type u_1

A, B: Set (Set α)

x: α

h.mp
x ∈ ⋂₀ (A ∪ B) → x ∈ ⋂₀ A ∩ ⋂₀ Bintro 
t: Set α
t
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ (A ∪ B)

this: ∀ (t : Set α), t ∈ A ∪ B → x ∈ t

t: Set α

h.mp
(t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)
    
α: Type u_1

A, B: Set (Set α)

x: α

h.mp
x ∈ ⋂₀ (A ∪ B) → x ∈ ⋂₀ A ∩ ⋂₀ Bexact ⟨
      fun 
ht: ?m.25445
ht => 
this: ∀ (t : Set α), t ∈ A ∪ B → x ∈ t
this 
t: Set α
t (
Set.mem_union_left: ∀ {α : Type ?u.25450} {x : α} {a : Set α} (b : Set α), x ∈ a → x ∈ a ∪ b
Set.mem_union_left 
B: Set (Set α)
B 
ht: ?m.25445
ht),
      fun 
ht: ?m.25468
ht => 
this: ∀ (t : Set α), t ∈ A ∪ B → x ∈ t
this 
t: Set α
t (
Set.mem_union_right: ∀ {α : Type ?u.25473} {x : α} {b : Set α} (a : Set α), x ∈ b → x ∈ a ∪ b
Set.mem_union_right 
A: Set (Set α)
A 
ht: ?m.25468
ht)
    ⟩
Goals accomplished! 🐙
  
α: Type u_1

A, B: Set (Set α)

⋂₀ (A ∪ B) = ⋂₀ A ∩ ⋂₀ B·
α: Type u_1

A, B: Set (Set α)

x: α

h.mpr
x ∈ ⋂₀ A ∩ ⋂₀ B → x ∈ ⋂₀ (A ∪ B) 
α: Type u_1

A, B: Set (Set α)

x: α

h.mpr
x ∈ ⋂₀ A ∩ ⋂₀ B → x ∈ ⋂₀ (A ∪ B)intro 
hx: ?b
hx
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

h.mpr
x ∈ ⋂₀ (A ∪ B)
    
α: Type u_1

A, B: Set (Set α)

x: α

h.mpr
x ∈ ⋂₀ A ∩ ⋂₀ B → x ∈ ⋂₀ (A ∪ B)have : ∀ 
t: Set α
t : 
Set: Type ?u.25486 → Type ?u.25486
Set 
α: Type u_1
α, (
t: Set α
t ∈ 
A: Set (Set α)
A → 
x: ?α
x ∈ 
t: Set α
t) ∧ (
t: Set α
t ∈ 
B: Set (Set α)
B → 
x: ?α
x ∈ 
t: Set α
t) := 
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

h.mpr
x ∈ ⋂₀ (A ∪ B)by
Goals accomplished! 🐙
      
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)have : (∀ 
t: Set α
t : 
Set: Type ?u.25567 → Type ?u.25567
Set 
α: Type u_1
α, 
t: Set α
t ∈ 
A: Set (Set α)
A → 
x: α
x ∈ 
t: Set α
t) ∧ (∀ 
t: Set α
t : 
Set: Type ?u.25608 → Type ?u.25608
Set 
α: Type u_1
α, 
t: Set α
t ∈ 
B: Set (Set α)
B → 
x: α
x ∈ 
t: Set α
t) := 
hx: x ∈ ⋂₀ A ∩ ⋂₀ B
hx
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

this: (∀ (t : Set α), t ∈ A → x ∈ t) ∧ ∀ (t : Set α), t ∈ B → x ∈ t

∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)
      
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)rwa [
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

this: (∀ (t : Set α), t ∈ A → x ∈ t) ∧ ∀ (t : Set α), t ∈ B → x ∈ t

∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)← 
forall_and: ∀ {α : Sort ?u.25652} {p q : α → Prop}, (∀ (x : α), p x ∧ q x) ↔ (∀ (x : α), p x) ∧ ∀ (x : α), q x
forall_and
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

this: ∀ (x_1 : Set α), (x_1 ∈ A → x ∈ x_1) ∧ (x_1 ∈ B → x ∈ x_1)

∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)]
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

this: ∀ (x_1 : Set α), (x_1 ∈ A → x ∈ x_1) ∧ (x_1 ∈ B → x ∈ x_1)

∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t) at 
this: (∀ (t : Set α), t ∈ A → x ∈ t) ∧ ∀ (t : Set α), t ∈ B → x ∈ t
this
Goals accomplished! 🐙
    
α: Type u_1

A, B: Set (Set α)

x: α

h.mpr
x ∈ ⋂₀ A ∩ ⋂₀ B → x ∈ ⋂₀ (A ∪ B)show ∀ (
t: Set α
t : 
Set: Type ?u.25695 → Type ?u.25695
Set 
α: Type u_1
α), 
t: Set α
t ∈ 
A: Set (Set α)
A ∪ 
B: Set (Set α)
B → 
x: ?α
x ∈ 
t: Set α
t
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

this: ∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)

h.mpr
∀ (t : Set α), t ∈ A ∪ B → x ∈ t
    
α: Type u_1

A, B: Set (Set α)

x: α

h.mpr
x ∈ ⋂₀ A ∩ ⋂₀ B → x ∈ ⋂₀ (A ∪ B)intro 
t: Set α
t 
ht: t ∈ A ∪ B
ht
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

this: ∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)

t: Set α

ht: t ∈ A ∪ B

h.mpr
x ∈ t
    
α: Type u_1

A, B: Set (Set α)

x: α

h.mpr
x ∈ ⋂₀ A ∩ ⋂₀ B → x ∈ ⋂₀ (A ∪ B)apply 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
ht: t ∈ A ∪ B
ht
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

this: ∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)

t: Set α

ht: t ∈ A ∪ B

h.mpr.left
t ∈ A → x ∈ t
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

this: ∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)

t: Set α

ht: t ∈ A ∪ B

h.mpr.right
t ∈ B → x ∈ t
    
α: Type u_1

A, B: Set (Set α)

x: α

h.mpr
x ∈ ⋂₀ A ∩ ⋂₀ B → x ∈ ⋂₀ (A ∪ B)·
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

this: ∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)

t: Set α

ht: t ∈ A ∪ B

h.mpr.left
t ∈ A → x ∈ t 
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

this: ∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)

t: Set α

ht: t ∈ A ∪ B

h.mpr.left
t ∈ A → x ∈ t
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

this: ∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)

t: Set α

ht: t ∈ A ∪ B

h.mpr.right
t ∈ B → x ∈ tintro 
hA: t ∈ A
hA
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

this: ∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)

t: Set α

ht: t ∈ A ∪ B

hA: t ∈ A

h.mpr.left
x ∈ t
      
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

this: ∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)

t: Set α

ht: t ∈ A ∪ B

h.mpr.left
t ∈ A → x ∈ texact (
this: ∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)
this 
t: Set α
t).
left: ∀ {a b : Prop}, a ∧ b → a
left 
hA: t ∈ A
hA
Goals accomplished! 🐙
    
α: Type u_1

A, B: Set (Set α)

x: α

h.mpr
x ∈ ⋂₀ A ∩ ⋂₀ B → x ∈ ⋂₀ (A ∪ B)·
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

this: ∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)

t: Set α

ht: t ∈ A ∪ B

h.mpr.right
t ∈ B → x ∈ t 
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

this: ∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)

t: Set α

ht: t ∈ A ∪ B

h.mpr.right
t ∈ B → x ∈ tintro 
hB: t ∈ B
hB
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

this: ∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)

t: Set α

ht: t ∈ A ∪ B

hB: t ∈ B

h.mpr.right
x ∈ t
      
α: Type u_1

A, B: Set (Set α)

x: α

hx: x ∈ ⋂₀ A ∩ ⋂₀ B

this: ∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)

t: Set α

ht: t ∈ A ∪ B

h.mpr.right
t ∈ B → x ∈ texact (
this: ∀ (t : Set α), (t ∈ A → x ∈ t) ∧ (t ∈ B → x ∈ t)
this 
t: Set α
t).
right: ∀ {a b : Prop}, a ∧ b → b
right 
hB: t ∈ B
hB
Goals accomplished! 🐙

/-- ### Exercise 4.24a

Show that is `𝓐` is nonempty, then `𝒫 (⋂ 𝓐) = ⋂ { 𝒫 X | X ∈ 𝓐 }`.
-/
theorem 
exercise_4_24a: ∀ {α : Type u_1} {𝓐 : Set (Set α)}, 𝒫⋂₀ 𝓐 = ⋂₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}
exercise_4_24a {
𝓐: Set (Set α)
𝓐 : 
Set: Type ?u.25861 → Type ?u.25861
Set (
Set: Type ?u.25862 → Type ?u.25862
Set 
α: ?m.25858
α)}
  : 𝒫 (⋂₀ 
𝓐: Set (Set α)
𝓐) = ⋂₀ { 𝒫 
X: ?m.25889
X | 
X: ?m.25889
X ∈ 
𝓐: Set (Set α)
𝓐 } := by
Goals accomplished! 🐙
  
α: Type u_1

𝓐: Set (Set α)

𝒫⋂₀ 𝓐 = ⋂₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}calc 𝒫 (⋂₀ 
𝓐: Set (Set α)
𝓐)
    
_: ?m✝
_ = { 
x: ?m.25985
x | 
x: ?m.25985
x ⊆ ⋂₀ 
𝓐: Set (Set α)
𝓐 } := 
rfl: ∀ {α : Sort ?u.26007} {a : α}, a = a
rfl
    
_: ?m✝
_ = { 
x: ?m.26087
x | 
x: ?m.26087
x ⊆ { 
y: ?m.26097
y | ∀ 
X: ?m.26100
X, 
X: ?m.26100
X ∈ 
𝓐: Set (Set α)
𝓐 → 
y: ?m.26097
y ∈ 
X: ?m.26100
X } } := 
rfl: ∀ {α : Sort ?u.26166} {a : α}, a = a
rfl
    
_: ?m✝
_ = { 
x: ?m.26241
x | ∀ 
t: ?m.26244
t ∈ 
x: ?m.26241
x, 
t: ?m.26244
t ∈ { 
y: ?m.26301
y | ∀ 
X: ?m.26304
X, 
X: ?m.26304
X ∈ 
𝓐: Set (Set α)
𝓐 → 
y: ?m.26301
y ∈ 
X: ?m.26304
X } } := 
rfl: ∀ {α : Sort ?u.26429} {a : α}, a = a
rfl
    
_: ?m✝
_ = { 
x: ?m.26494
x | ∀ 
t: ?m.26497
t ∈ 
x: ?m.26494
x, (∀ 
X: ?m.26536
X, 
X: ?m.26536
X ∈ 
𝓐: Set (Set α)
𝓐 → 
t: ?m.26497
t ∈ 
X: ?m.26536
X) } := 
rfl: ∀ {α : Sort ?u.26642} {a : α}, a = a
rfl
    
_: ?m✝
_ = { 
x: ?m.26714
x | ∀ 
X: ?m.26717
X ∈ 
𝓐: Set (Set α)
𝓐, (∀ 
t: ?m.26747
t, 
t: ?m.26747
t ∈ 
x: ?m.26714
x → 
t: ?m.26747
t ∈ 
X: ?m.26717
X) } := 
α: Type u_1

𝓐: Set (Set α)

𝒫⋂₀ 𝓐 = ⋂₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}by
Goals accomplished! 🐙
      
α: Type u_1

𝓐: Set (Set α)

{x | ∀ (t : α), t ∈ x → ∀ (X : Set α), X ∈ 𝓐 → t ∈ X} = {x | ∀ (X : Set α), X ∈ 𝓐 → ∀ (t : α), t ∈ x → t ∈ X}ext
α: Type u_1

𝓐: Set (Set α)

x✝: Set α

h
x✝ ∈ {x | ∀ (t : α), t ∈ x → ∀ (X : Set α), X ∈ 𝓐 → t ∈ X} ↔ x✝ ∈ {x | ∀ (X : Set α), X ∈ 𝓐 → ∀ (t : α), t ∈ x → t ∈ X}
      
α: Type u_1

𝓐: Set (Set α)

{x | ∀ (t : α), t ∈ x → ∀ (X : Set α), X ∈ 𝓐 → t ∈ X} = {x | ∀ (X : Set α), X ∈ 𝓐 → ∀ (t : α), t ∈ x → t ∈ X}rw [
α: Type u_1

𝓐: Set (Set α)

x✝: Set α

h
x✝ ∈ {x | ∀ (t : α), t ∈ x → ∀ (X : Set α), X ∈ 𝓐 → t ∈ X} ↔ x✝ ∈ {x | ∀ (X : Set α), X ∈ 𝓐 → ∀ (t : α), t ∈ x → t ∈ X}
Set.mem_setOf: ∀ {α : Type ?u.27490} {a : α} {p : α → Prop}, a ∈ {x | p x} ↔ p a
Set.mem_setOf,
α: Type u_1

𝓐: Set (Set α)

x✝: Set α

h
(∀ (t : α), t ∈ x✝ → ∀ (X : Set α), X ∈ 𝓐 → t ∈ X) ↔ x✝ ∈ {x | ∀ (X : Set α), X ∈ 𝓐 → ∀ (t : α), t ∈ x → t ∈ X} 
α: Type u_1

𝓐: Set (Set α)

x✝: Set α

h
x✝ ∈ {x | ∀ (t : α), t ∈ x → ∀ (X : Set α), X ∈ 𝓐 → t ∈ X} ↔ x✝ ∈ {x | ∀ (X : Set α), X ∈ 𝓐 → ∀ (t : α), t ∈ x → t ∈ X}
Set.mem_setOf: ∀ {α : Type ?u.27533} {a : α} {p : α → Prop}, a ∈ {x | p x} ↔ p a
Set.mem_setOf,
α: Type u_1

𝓐: Set (Set α)

x✝: Set α

h
(∀ (t : α), t ∈ x✝ → ∀ (X : Set α), X ∈ 𝓐 → t ∈ X) ↔ ∀ (X : Set α), X ∈ 𝓐 → ∀ (t : α), t ∈ x✝ → t ∈ X 
α: Type u_1

𝓐: Set (Set α)

x✝: Set α

h
x✝ ∈ {x | ∀ (t : α), t ∈ x → ∀ (X : Set α), X ∈ 𝓐 → t ∈ X} ↔ x✝ ∈ {x | ∀ (X : Set α), X ∈ 𝓐 → ∀ (t : α), t ∈ x → t ∈ X}
forall_mem_comm: ∀ {α : Type ?u.27555} {β : Type ?u.27556} {X : Set α} {Y : Set β} (p : α → β → Prop),
  (∀ (u : α), u ∈ X → ∀ (v : β), v ∈ Y → p u v) = ∀ (v : β), v ∈ Y → ∀ (u : α), u ∈ X → p u v
forall_mem_comm 
(· ∈ ·): α → Set α → Prop
(· ∈ ·)
α: Type u_1

𝓐: Set (Set α)

x✝: Set α

h
(∀ (v : Set α), v ∈ 𝓐 → ∀ (u : α), u ∈ x✝ → u ∈ v) ↔ ∀ (X : Set α), X ∈ 𝓐 → ∀ (t : α), t ∈ x✝ → t ∈ X]
Goals accomplished! 🐙
    
_: ?m✝
_ = { 
x: ?m.26928
x | ∀ 
X: ?m.26931
X ∈ 
𝓐: Set (Set α)
𝓐, 
x: ?m.26928
x ⊆ 
X: ?m.26931
X} := 
rfl: ∀ {α : Sort ?u.26977} {a : α}, a = a
rfl
    
_: ?m✝
_ = { 
x: ?m.27046
x | ∀ 
X: ?m.27049
X ∈ 
𝓐: Set (Set α)
𝓐, 
x: ?m.27046
x ∈ 𝒫 
X: ?m.27049
X } := 
rfl: ∀ {α : Sort ?u.27115} {a : α}, a = a
rfl
    
_: ?m✝
_ = { 
x: ?m.27184
x | ∀ 
t: ?m.27187
t ∈ { 𝒫 
X: ?m.27215
X | 
X: ?m.27215
X ∈ 
𝓐: Set (Set α)
𝓐 }, 
x: ?m.27184
x ∈ 
t: ?m.27187
t} := 
α: Type u_1

𝓐: Set (Set α)

𝒫⋂₀ 𝓐 = ⋂₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}by
Goals accomplished! 🐙 
α: Type u_1

𝓐: Set (Set α)

{x | ∀ (X : Set α), X ∈ 𝓐 → x ∈ 𝒫 X} = {x | ∀ (t : Set (Set α)), t ∈ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} → x ∈ t}simp
Goals accomplished! 🐙
    
_: ?m✝
_ = ⋂₀ { 𝒫 
X: ?m.27354
X | 
X: ?m.27354
X ∈ 
𝓐: Set (Set α)
𝓐 } := 
rfl: ∀ {α : Sort ?u.27411} {a : α}, a = a
rfl
Goals accomplished! 🐙

/-- ### Exercise 4.24b

Show that
```
⋃ {𝒫 X | X ∈ 𝓐} ⊆ 𝒫 ⋃ 𝓐.
```
Under what conditions does equality hold?
-/
theorem 
exercise_4_24b: ∀ {α : Type u_1} {𝓐 : Set (Set α)},
  ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐 ∧ (⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐 ↔ ⋃₀ 𝓐 ∈ 𝓐)
exercise_4_24b {
𝓐: Set (Set α)
𝓐 : 
Set: Type ?u.33730 → Type ?u.33730
Set (
Set: Type ?u.33731 → Type ?u.33731
Set 
α: ?m.33727
α)}
  : (⋃₀ { 𝒫 
X: ?m.33750
X | 
X: ?m.33750
X ∈ 
𝓐: Set (Set α)
𝓐 } ⊆ 𝒫 ⋃₀ 
𝓐: Set (Set α)
𝓐)
  ∧ ((⋃₀ { 𝒫 
X: ?m.33837
X | 
X: ?m.33837
X ∈ 
𝓐: Set (Set α)
𝓐 } = 𝒫 ⋃₀ 
𝓐: Set (Set α)
𝓐) ↔ (⋃₀ 
𝓐: Set (Set α)
𝓐 ∈ 
𝓐: Set (Set α)
𝓐)) := by
Goals accomplished! 🐙
  
α: Type u_1

𝓐: Set (Set α)

⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐 ∧ (⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐 ↔ ⋃₀ 𝓐 ∈ 𝓐)have 
hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐
hS : (⋃₀ { 𝒫 
X: ?m.33949
X | 
X: ?m.33949
X ∈ 
𝓐: Set (Set α)
𝓐 } ⊆ 𝒫 ⋃₀ 
𝓐: Set (Set α)
𝓐) := 
α: Type u_1

𝓐: Set (Set α)

⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐 ∧ (⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐 ↔ ⋃₀ 𝓐 ∈ 𝓐)by
Goals accomplished! 🐙
    
α: Type u_1

𝓐: Set (Set α)

⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐simp
α: Type u_1

𝓐: Set (Set α)

∀ (a : Set α), a ∈ 𝓐 → a ⊆ ⋃₀ 𝓐
    
α: Type u_1

𝓐: Set (Set α)

⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐exact 
exercise_3_3: ∀ {α : Type ?u.37198} {A : Set (Set α)} (x : Set α), x ∈ A → x ⊆ ⋃₀ A
exercise_3_3
Goals accomplished! 🐙
  
α: Type u_1

𝓐: Set (Set α)

⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐 ∧ (⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐 ↔ ⋃₀ 𝓐 ∈ 𝓐)refine ⟨
hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐
hS, 
?_: ?m.37218
?_⟩
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐 ↔ ⋃₀ 𝓐 ∈ 𝓐
  
α: Type u_1

𝓐: Set (Set α)

⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐 ∧ (⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐 ↔ ⋃₀ 𝓐 ∈ 𝓐)apply 
Iff.intro: ∀ {a b : Prop}, (a → b) → (b → a) → (a ↔ b)
Iff.intro
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

mp
⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐 → ⋃₀ 𝓐 ∈ 𝓐
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

mpr
⋃₀ 𝓐 ∈ 𝓐 → ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐
  
α: Type u_1

𝓐: Set (Set α)

⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐 ∧ (⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐 ↔ ⋃₀ 𝓐 ∈ 𝓐)·
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

mp
⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐 → ⋃₀ 𝓐 ∈ 𝓐 
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

mp
⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐 → ⋃₀ 𝓐 ∈ 𝓐
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

mpr
⋃₀ 𝓐 ∈ 𝓐 → ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐intro 
rS: ?a
rS
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

mp
⋃₀ 𝓐 ∈ 𝓐
    
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

mp
⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐 → ⋃₀ 𝓐 ∈ 𝓐have 
rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}
rS : 𝒫 ⋃₀ 
𝓐: Set (Set α)
𝓐 ⊆ ⋃₀ { 𝒫 
X: ?m.37267
X | 
X: ?m.37267
X ∈ 
𝓐: Set (Set α)
𝓐 } :=
      (
Set.Subset.antisymm_iff: ∀ {α : Type ?u.37297} {a b : Set α}, a = b ↔ a ⊆ b ∧ b ⊆ a
Set.Subset.antisymm_iff.
mp: ∀ {a b : Prop}, (a ↔ b) → a → b
mp 
rS: ?a
rS).
right: ∀ {a b : Prop}, a ∧ b → b
right
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

mp
⋃₀ 𝓐 ∈ 𝓐
    
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

mp
⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐 → ⋃₀ 𝓐 ∈ 𝓐have 
hA: ⋃₀ 𝓐 ∈ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}
hA : ⋃₀ 
𝓐: Set (Set α)
𝓐 ∈ ⋃₀ { 𝒫 
X: ?m.37348
X | 
X: ?m.37348
X ∈ 
𝓐: Set (Set α)
𝓐 } :=
      
rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}
rS 
Set.self_mem_powerset_self: ∀ {α : Type ?u.37411} {A : Set α}, A ∈ 𝒫 A
Set.self_mem_powerset_self
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

mp
⋃₀ 𝓐 ∈ 𝓐
    
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

mp
⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐 → ⋃₀ 𝓐 ∈ 𝓐conv at hA =>
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

{a | ∃ t, t ∈ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ∧ a ∈ t}
      
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

mp
⋃₀ 𝓐 ∈ 𝓐rhs
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}
      
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

⋃₀ 𝓐 ∈ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}unfold 
Set.sUnion: {α : Type u_1} → Set (Set α) → Set α
Set.sUnion 
sSup: {α : Type u_1} → [self : SupSet α] → Set α → α
sSup 
Set.instSupSetSet: {α : Type u_1} → SupSet (Set α)
Set.instSupSetSet
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

{ sSup := fun s => {a | ∃ t, t ∈ s ∧ a ∈ t} }.1 {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}
      
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

⋃₀ 𝓐 ∈ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}simp only
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

{a | ∃ t, t ∈ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ∧ a ∈ t}
    
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

mp
⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐 → ⋃₀ 𝓐 ∈ 𝓐have ⟨
X: Set (Set α)
X, ⟨⟨
x: Set α
x, 
hx: x ∈ 𝓐 ∧ 𝒫 x = X
hx⟩, 
ht: ⋃₀ 𝓐 ∈ X
ht⟩⟩ := 
Set.mem_setOf: ∀ {α : Type ?u.37549} {a : α} {p : α → Prop}, a ∈ {x | p x} ↔ p a
Set.mem_setOf.
mp: ∀ {a b : Prop}, (a ↔ b) → a → b
mp 
hA: ⋃₀ 𝓐 ∈ {a | ∃ t, t ∈ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ∧ a ∈ t}
hA
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ {a | ∃ t, t ∈ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ∧ a ∈ t}

X: Set (Set α)

x: Set α

hx: x ∈ 𝓐 ∧ 𝒫 x = X

ht: ⋃₀ 𝓐 ∈ X

mp
⋃₀ 𝓐 ∈ 𝓐
    
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

mp
⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐 → ⋃₀ 𝓐 ∈ 𝓐have : ⋃₀ 
𝓐: Set (Set α)
𝓐 = 
x: Set α
x := 
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ {a | ∃ t, t ∈ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ∧ a ∈ t}

X: Set (Set α)

x: Set α

hx: x ∈ 𝓐 ∧ 𝒫 x = X

ht: ⋃₀ 𝓐 ∈ X

mp
⋃₀ 𝓐 ∈ 𝓐by
Goals accomplished! 🐙
      
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ {a | ∃ t, t ∈ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ∧ a ∈ t}

X: Set (Set α)

x: Set α

hx: x ∈ 𝓐 ∧ 𝒫 x = X

ht: ⋃₀ 𝓐 ∈ X

⋃₀ 𝓐 = xrw [
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ {a | ∃ t, t ∈ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ∧ a ∈ t}

X: Set (Set α)

x: Set α

hx: x ∈ 𝓐 ∧ 𝒫 x = X

ht: ⋃₀ 𝓐 ∈ X

⋃₀ 𝓐 = x← 
hx: x ∈ 𝓐 ∧ 𝒫 x = X
hx.
right: ∀ {a b : Prop}, a ∧ b → b
right
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ {a | ∃ t, t ∈ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ∧ a ∈ t}

X: Set (Set α)

x: Set α

hx: x ∈ 𝓐 ∧ 𝒫 x = X

ht: ⋃₀ 𝓐 ∈ 𝒫 x

⋃₀ 𝓐 = x]
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ {a | ∃ t, t ∈ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ∧ a ∈ t}

X: Set (Set α)

x: Set α

hx: x ∈ 𝓐 ∧ 𝒫 x = X

ht: ⋃₀ 𝓐 ∈ 𝒫 x

⋃₀ 𝓐 = x at 
ht: ⋃₀ 𝓐 ∈ X
ht
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ {a | ∃ t, t ∈ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ∧ a ∈ t}

X: Set (Set α)

x: Set α

hx: x ∈ 𝓐 ∧ 𝒫 x = X

ht: ⋃₀ 𝓐 ∈ 𝒫 x

⋃₀ 𝓐 = x
      
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ {a | ∃ t, t ∈ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ∧ a ∈ t}

X: Set (Set α)

x: Set α

hx: x ∈ 𝓐 ∧ 𝒫 x = X

ht: ⋃₀ 𝓐 ∈ X

⋃₀ 𝓐 = xhave 
hl: ⋃₀ 𝓐 ⊆ x
hl : ⋃₀ 
𝓐: Set (Set α)
𝓐 ⊆ 
x: Set α
x := 
ht: ⋃₀ 𝓐 ∈ 𝒫 x
ht
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ {a | ∃ t, t ∈ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ∧ a ∈ t}

X: Set (Set α)

x: Set α

hx: x ∈ 𝓐 ∧ 𝒫 x = X

ht: ⋃₀ 𝓐 ∈ 𝒫 x

hl: ⋃₀ 𝓐 ⊆ x

⋃₀ 𝓐 = x
      
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ {a | ∃ t, t ∈ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ∧ a ∈ t}

X: Set (Set α)

x: Set α

hx: x ∈ 𝓐 ∧ 𝒫 x = X

ht: ⋃₀ 𝓐 ∈ X

⋃₀ 𝓐 = xhave 
hr: x ⊆ ⋃₀ 𝓐
hr : 
x: Set α
x ⊆ ⋃₀ 
𝓐: Set (Set α)
𝓐 := 
exercise_3_3: ∀ {α : Type ?u.37856} {A : Set (Set α)} (x : Set α), x ∈ A → x ⊆ ⋃₀ A
exercise_3_3 
x: Set α
x 
hx: x ∈ 𝓐 ∧ 𝒫 x = X
hx.
left: ∀ {a b : Prop}, a ∧ b → a
left
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ {a | ∃ t, t ∈ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ∧ a ∈ t}

X: Set (Set α)

x: Set α

hx: x ∈ 𝓐 ∧ 𝒫 x = X

ht: ⋃₀ 𝓐 ∈ 𝒫 x

hl: ⋃₀ 𝓐 ⊆ x

hr: x ⊆ ⋃₀ 𝓐

⋃₀ 𝓐 = x
      
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ {a | ∃ t, t ∈ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ∧ a ∈ t}

X: Set (Set α)

x: Set α

hx: x ∈ 𝓐 ∧ 𝒫 x = X

ht: ⋃₀ 𝓐 ∈ X

⋃₀ 𝓐 = xexact 
Set.Subset.antisymm: ∀ {α : Type ?u.37876} {a b : Set α}, a ⊆ b → b ⊆ a → a = b
Set.Subset.antisymm 
hl: ⋃₀ 𝓐 ⊆ x
hl 
hr: x ⊆ ⋃₀ 𝓐
hr
Goals accomplished! 🐙
    
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

mp
⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐 → ⋃₀ 𝓐 ∈ 𝓐rw [
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ {a | ∃ t, t ∈ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ∧ a ∈ t}

X: Set (Set α)

x: Set α

hx: x ∈ 𝓐 ∧ 𝒫 x = X

ht: ⋃₀ 𝓐 ∈ X

this: ⋃₀ 𝓐 = x

mp
⋃₀ 𝓐 ∈ 𝓐← 
this: ⋃₀ 𝓐 = x
this
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ {a | ∃ t, t ∈ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ∧ a ∈ t}

X: Set (Set α)

x: Set α

hx: ⋃₀ 𝓐 ∈ 𝓐 ∧ 𝒫⋃₀ 𝓐 = X

ht: ⋃₀ 𝓐 ∈ X

this: ⋃₀ 𝓐 = x

mp
⋃₀ 𝓐 ∈ 𝓐]
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ {a | ∃ t, t ∈ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ∧ a ∈ t}

X: Set (Set α)

x: Set α

hx: ⋃₀ 𝓐 ∈ 𝓐 ∧ 𝒫⋃₀ 𝓐 = X

ht: ⋃₀ 𝓐 ∈ X

this: ⋃₀ 𝓐 = x

mp
⋃₀ 𝓐 ∈ 𝓐 at 
hx: x ∈ 𝓐 ∧ 𝒫 x = X
hx
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

rS✝: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐

rS: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}

hA: ⋃₀ 𝓐 ∈ {a | ∃ t, t ∈ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ∧ a ∈ t}

X: Set (Set α)

x: Set α

hx: ⋃₀ 𝓐 ∈ 𝓐 ∧ 𝒫⋃₀ 𝓐 = X

ht: ⋃₀ 𝓐 ∈ X

this: ⋃₀ 𝓐 = x

mp
⋃₀ 𝓐 ∈ 𝓐
    
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

mp
⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐 → ⋃₀ 𝓐 ∈ 𝓐exact 
hx: ⋃₀ 𝓐 ∈ 𝓐 ∧ 𝒫⋃₀ 𝓐 = X
hx.
left: ∀ {a b : Prop}, a ∧ b → a
left
Goals accomplished! 🐙
  
α: Type u_1

𝓐: Set (Set α)

⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐 ∧ (⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐 ↔ ⋃₀ 𝓐 ∈ 𝓐)·
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

mpr
⋃₀ 𝓐 ∈ 𝓐 → ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐 
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

mpr
⋃₀ 𝓐 ∈ 𝓐 → ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐intro 
hA: ?b
hA
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

hA: ⋃₀ 𝓐 ∈ 𝓐

mpr
⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐
    
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

mpr
⋃₀ 𝓐 ∈ 𝓐 → ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐suffices 𝒫 ⋃₀ 
𝓐: Set (Set α)
𝓐 ⊆ ⋃₀ { 𝒫 
X: ?m.37944
X | 
X: ?m.37944
X ∈ 
𝓐: Set (Set α)
𝓐 } from
      
subset_antisymm: ∀ {α : Type ?u.37981} [inst : HasSubset α] {a b : α} [inst_1 : IsAntisymm α fun x x_1 => x ⊆ x_1], a ⊆ b → b ⊆ a → a = b
subset_antisymm 
hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐
hS 
this: 𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}
this
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

hA: ⋃₀ 𝓐 ∈ 𝓐

mpr
𝒫⋃₀ 𝓐 ⊆ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}
    
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

mpr
⋃₀ 𝓐 ∈ 𝓐 → ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐show ∀ 
x: ?m.38021
x, 
x: ?m.38021
x ∈ 𝒫 ⋃₀ 
𝓐: Set (Set α)
𝓐 → 
x: ?m.38021
x ∈ ⋃₀ { 
x: ?m.38084
x | ∃ 
X: ?m.38089
X, 
X: ?m.38089
X ∈ 
𝓐: Set (Set α)
𝓐 ∧ 𝒫 
X: ?m.38089
X = 
x: ?m.38084
x }
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

hA: ⋃₀ 𝓐 ∈ 𝓐

mpr
∀ (x : Set α), x ∈ 𝒫⋃₀ 𝓐 → x ∈ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}
    
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

mpr
⋃₀ 𝓐 ∈ 𝓐 → ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐intro 
x: Set α
x 
hx: x ∈ 𝒫⋃₀ 𝓐
hx
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

hA: ⋃₀ 𝓐 ∈ 𝓐

x: Set α

hx: x ∈ 𝒫⋃₀ 𝓐

mpr
x ∈ ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}
    
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

mpr
⋃₀ 𝓐 ∈ 𝓐 → ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐unfold 
Set.sUnion: {α : Type u_1} → Set (Set α) → Set α
Set.sUnion 
sSup: {α : Type u_1} → [self : SupSet α] → Set α → α
sSup 
Set.instSupSetSet: {α : Type u_1} → SupSet (Set α)
Set.instSupSetSet
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

hA: ⋃₀ 𝓐 ∈ 𝓐

x: Set α

hx: x ∈ 𝒫⋃₀ 𝓐

mpr
x ∈ { sSup := fun s => {a | ∃ t, t ∈ s ∧ a ∈ t} }.1 {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x}
    
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

mpr
⋃₀ 𝓐 ∈ 𝓐 → ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐simp only [
Set.mem_setOf_eq: ∀ {α : Type ?u.38187} {x : α} {p : α → Prop}, (x ∈ {y | p y}) = p x
Set.mem_setOf_eq, 
exists_exists_and_eq_and: ∀ {α : Sort ?u.38205} {β : Sort ?u.38204} {f : α → β} {p : α → Prop} {q : β → Prop},
  (∃ b, (∃ a, p a ∧ f a = b) ∧ q b) ↔ ∃ a, p a ∧ q (f a)
exists_exists_and_eq_and, 
Set.mem_powerset_iff: ∀ {α : Type ?u.38227} (x s : Set α), x ∈ 𝒫 s ↔ x ⊆ s
Set.mem_powerset_iff]
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

hA: ⋃₀ 𝓐 ∈ 𝓐

x: Set α

hx: x ∈ 𝒫⋃₀ 𝓐

mpr
∃ a, a ∈ 𝓐 ∧ x ⊆ a
    
α: Type u_1

𝓐: Set (Set α)

hS: ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} ⊆ 𝒫⋃₀ 𝓐

mpr
⋃₀ 𝓐 ∈ 𝓐 → ⋃₀ {x | ∃ X, X ∈ 𝓐 ∧ 𝒫 X = x} = 𝒫⋃₀ 𝓐exact ⟨⋃₀ 
𝓐: Set (Set α)
𝓐, ⟨
hA: ?b
hA, 
hx: x ∈ 𝒫⋃₀ 𝓐
hx⟩⟩
Goals accomplished! 🐙

/-- ### Exercise 4.25

Is `A ∪ (⋃ 𝓑)` always the same as `⋃ { A ∪ X | X ∈ 𝓑 }`? If not, then under
what conditions does equality hold? 
-/
theorem 
exercise_4_25: ∀ {α : Type u_1} {A : Set α} (𝓑 : Set (Set α)), A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x} ↔ A = ∅ ∨ Set.Nonempty 𝓑
exercise_4_25 {
A: Set α
A : 
Set: Type ?u.38862 → Type ?u.38862
Set 
α: ?m.38859
α} (
𝓑: Set (Set α)
𝓑 : 
Set: Type ?u.38865 → Type ?u.38865
Set (
Set: Type ?u.38866 → Type ?u.38866
Set 
α: ?m.38859
α))
  : (
A: Set α
A ∪ (⋃₀ 
𝓑: Set (Set α)
𝓑) = ⋃₀ { 
A: Set α
A ∪ 
X: ?m.38899
X | 
X: ?m.38899
X ∈ 
𝓑: Set (Set α)
𝓑 }) ↔ (
A: Set α
A = 
∅: ?m.38962
∅ ∨ 
Set.Nonempty: {α : Type ?u.39037} → Set α → Prop
Set.Nonempty 
𝓑: Set (Set α)
𝓑) := by
Goals accomplished! 🐙
  
α: Type u_1

A: Set α

𝓑: Set (Set α)

A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x} ↔ A = ∅ ∨ Set.Nonempty 𝓑apply 
Iff.intro: ∀ {a b : Prop}, (a → b) → (b → a) → (a ↔ b)
Iff.intro
α: Type u_1

A: Set α

𝓑: Set (Set α)

mp
A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x} → A = ∅ ∨ Set.Nonempty 𝓑
α: Type u_1

A: Set α

𝓑: Set (Set α)

mpr
A = ∅ ∨ Set.Nonempty 𝓑 → A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}
  
α: Type u_1

A: Set α

𝓑: Set (Set α)

A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x} ↔ A = ∅ ∨ Set.Nonempty 𝓑·
α: Type u_1

A: Set α

𝓑: Set (Set α)

mp
A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x} → A = ∅ ∨ Set.Nonempty 𝓑 
α: Type u_1

A: Set α

𝓑: Set (Set α)

mp
A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x} → A = ∅ ∨ Set.Nonempty 𝓑
α: Type u_1

A: Set α

𝓑: Set (Set α)

mpr
A = ∅ ∨ Set.Nonempty 𝓑 → A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}intro 
h: ?a
h
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}

mp
A = ∅ ∨ Set.Nonempty 𝓑
    
α: Type u_1

A: Set α

𝓑: Set (Set α)

mp
A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x} → A = ∅ ∨ Set.Nonempty 𝓑by_cases 
h𝓑: ?m.39081
h𝓑 : 
Set.Nonempty: {α : Type ?u.39060} → Set α → Prop
Set.Nonempty 
𝓑: Set (Set α)
𝓑
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}

h𝓑: Set.Nonempty 𝓑

pos
A = ∅ ∨ Set.Nonempty 𝓑
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}

h𝓑: ¬Set.Nonempty 𝓑

neg
A = ∅ ∨ Set.Nonempty 𝓑
    
α: Type u_1

A: Set α

𝓑: Set (Set α)

mp
A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x} → A = ∅ ∨ Set.Nonempty 𝓑·
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}

h𝓑: Set.Nonempty 𝓑

pos
A = ∅ ∨ Set.Nonempty 𝓑 
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}

h𝓑: Set.Nonempty 𝓑

pos
A = ∅ ∨ Set.Nonempty 𝓑
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}

h𝓑: ¬Set.Nonempty 𝓑

neg
A = ∅ ∨ Set.Nonempty 𝓑exact 
Or.inr: ∀ {a b : Prop}, b → a ∨ b
Or.inr 
h𝓑: ?m.39074
h𝓑
Goals accomplished! 🐙
    
α: Type u_1

A: Set α

𝓑: Set (Set α)

mp
A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x} → A = ∅ ∨ Set.Nonempty 𝓑·
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}

h𝓑: ¬Set.Nonempty 𝓑

neg
A = ∅ ∨ Set.Nonempty 𝓑 
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}

h𝓑: ¬Set.Nonempty 𝓑

neg
A = ∅ ∨ Set.Nonempty 𝓑have : 
𝓑: Set (Set α)
𝓑 = 
∅: ?m.39096
∅ := 
Set.not_nonempty_iff_eq_empty: ∀ {α : Type ?u.39172} {s : Set α}, ¬Set.Nonempty s ↔ s = ∅
Set.not_nonempty_iff_eq_empty.
mp: ∀ {a b : Prop}, (a ↔ b) → a → b
mp 
h𝓑: ?m.39081
h𝓑
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}

h𝓑: ¬Set.Nonempty 𝓑

this: 𝓑 = ∅

neg
A = ∅ ∨ Set.Nonempty 𝓑
      
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}

h𝓑: ¬Set.Nonempty 𝓑

neg
A = ∅ ∨ Set.Nonempty 𝓑rw [
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}

h𝓑: ¬Set.Nonempty 𝓑

this: 𝓑 = ∅

neg
A = ∅ ∨ Set.Nonempty 𝓑
this: 𝓑 = ∅
this
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A ∪ ⋃₀ ∅ = ⋃₀ {x | ∃ X, X ∈ ∅ ∧ A ∪ X = x}

h𝓑: ¬Set.Nonempty 𝓑

this: 𝓑 = ∅

neg
A = ∅ ∨ Set.Nonempty 𝓑]
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A ∪ ⋃₀ ∅ = ⋃₀ {x | ∃ X, X ∈ ∅ ∧ A ∪ X = x}

h𝓑: ¬Set.Nonempty 𝓑

this: 𝓑 = ∅

neg
A = ∅ ∨ Set.Nonempty 𝓑 at 
h: ?a
h
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A ∪ ⋃₀ ∅ = ⋃₀ {x | ∃ X, X ∈ ∅ ∧ A ∪ X = x}

h𝓑: ¬Set.Nonempty 𝓑

this: 𝓑 = ∅

neg
A = ∅ ∨ Set.Nonempty 𝓑
      
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}

h𝓑: ¬Set.Nonempty 𝓑

neg
A = ∅ ∨ Set.Nonempty 𝓑simp at 
h: A ∪ ⋃₀ ∅ = ⋃₀ {x | ∃ X, X ∈ ∅ ∧ A ∪ X = x}
h
α: Type u_1

A: Set α

𝓑: Set (Set α)

h𝓑: ¬Set.Nonempty 𝓑

this: 𝓑 = ∅

h: A = ∅

neg
A = ∅ ∨ Set.Nonempty 𝓑
      
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}

h𝓑: ¬Set.Nonempty 𝓑

neg
A = ∅ ∨ Set.Nonempty 𝓑exact 
Or.inl: ∀ {a b : Prop}, a → a ∨ b
Or.inl 
h: A = ∅
h
Goals accomplished! 🐙
  
α: Type u_1

A: Set α

𝓑: Set (Set α)

A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x} ↔ A = ∅ ∨ Set.Nonempty 𝓑·
α: Type u_1

A: Set α

𝓑: Set (Set α)

mpr
A = ∅ ∨ Set.Nonempty 𝓑 → A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x} 
α: Type u_1

A: Set α

𝓑: Set (Set α)

mpr
A = ∅ ∨ Set.Nonempty 𝓑 → A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}intro 
h: ?b
h
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

mpr
A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}
    
α: Type u_1

A: Set α

𝓑: Set (Set α)

mpr
A = ∅ ∨ Set.Nonempty 𝓑 → A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}apply 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
h: ?b
h
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

mpr.left
A = ∅ → A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

mpr.right
Set.Nonempty 𝓑 → A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}
    
α: Type u_1

A: Set α

𝓑: Set (Set α)

mpr
A = ∅ ∨ Set.Nonempty 𝓑 → A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}·
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

mpr.left
A = ∅ → A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x} 
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

mpr.left
A = ∅ → A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

mpr.right
Set.Nonempty 𝓑 → A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}intro 
hA: A = ∅
hA
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

hA: A = ∅

mpr.left
A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}
      
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

mpr.left
A = ∅ → A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}rw [
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

hA: A = ∅

mpr.left
A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}
hA: A = ∅
hA
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

hA: A = ∅

mpr.left
∅ ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ ∅ ∪ X = x}]
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

hA: A = ∅

mpr.left
∅ ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ ∅ ∪ X = x}
      
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

mpr.left
A = ∅ → A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}simp
Goals accomplished! 🐙
    
α: Type u_1

A: Set α

𝓑: Set (Set α)

mpr
A = ∅ ∨ Set.Nonempty 𝓑 → A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}·
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

mpr.right
Set.Nonempty 𝓑 → A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x} 
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

mpr.right
Set.Nonempty 𝓑 → A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}intro 
h𝓑: Set.Nonempty 𝓑
h𝓑
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

mpr.right
A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}
      
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

mpr.right
Set.Nonempty 𝓑 → A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}calc 
A: Set α
A ∪ (⋃₀ 
𝓑: Set (Set α)
𝓑)
        
_: ?m✝
_ = { 
x: ?m.41070
x | 
x: ?m.41070
x ∈ 
A: Set α
A ∨ 
x: ?m.41070
x ∈ ⋃₀ 
𝓑: Set (Set α)
𝓑} := 
rfl: ∀ {α : Sort ?u.41109} {a : α}, a = a
rfl
        
_: ?m✝
_ = { 
x: ?m.41189
x | 
x: ?m.41189
x ∈ 
A: Set α
A ∨ (∃ 
b: ?m.41208
b ∈ 
𝓑: Set (Set α)
𝓑, 
x: ?m.41189
x ∈ 
b: ?m.41208
b) } := 
rfl: ∀ {α : Sort ?u.41236} {a : α}, a = a
rfl
        
_: ?m✝
_ = { 
x: ?m.41304
x | ∃ 
b: ?m.41309
b ∈ 
𝓑: Set (Set α)
𝓑, 
x: ?m.41304
x ∈ 
A: Set α
A ∨ 
x: ?m.41304
x ∈ 
b: ?m.41309
b } := 
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

mpr.right
A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}by
Goals accomplished! 🐙
          
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

{x | x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b} = {x | ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)}ext 
x: ?α
x
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

h
x ∈ {x | x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b} ↔ x ∈ {x | ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)}
          
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

{x | x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b} = {x | ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)}rw [
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

h
x ∈ {x | x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b} ↔ x ∈ {x | ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)}
Set.mem_setOf: ∀ {α : Type ?u.41820} {a : α} {p : α → Prop}, a ∈ {x | p x} ↔ p a
Set.mem_setOf,
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

h
(x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b) ↔ x ∈ {x | ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)} 
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

h
x ∈ {x | x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b} ↔ x ∈ {x | ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)}
Set.mem_setOf: ∀ {α : Type ?u.41838} {a : α} {p : α → Prop}, a ∈ {x | p x} ↔ p a
Set.mem_setOf
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

h
(x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b) ↔ ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)]
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

h
(x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b) ↔ ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)
          
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

{x | x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b} = {x | ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)}apply 
Iff.intro: ∀ {a b : Prop}, (a → b) → (b → a) → (a ↔ b)
Iff.intro
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

h.mp
(x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b) → ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

h.mpr
(∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)) → x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b
          
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

{x | x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b} = {x | ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)}·
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

h.mp
(x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b) → ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b) 
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

h.mp
(x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b) → ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

h.mpr
(∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)) → x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ bintro 
hx: ?a
hx
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

hx: x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b

h.mp
∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)
            
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

h.mp
(x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b) → ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)apply 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
hx: ?a
hx
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

hx: x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b

h.mp.left
x ∈ A → ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

hx: x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b

h.mp.right
(∃ b, b ∈ 𝓑 ∧ x ∈ b) → ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)
            
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

h.mp
(x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b) → ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)·
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

hx: x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b

h.mp.left
x ∈ A → ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b) 
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

hx: x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b

h.mp.left
x ∈ A → ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

hx: x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b

h.mp.right
(∃ b, b ∈ 𝓑 ∧ x ∈ b) → ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)intro 
hA: x ∈ A
hA
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

hx: x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b

hA: x ∈ A

h.mp.left
∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)
              
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

hx: x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b

h.mp.left
x ∈ A → ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)have ⟨
b: Set α
b, 
hb: b ∈ 𝓑
hb⟩ := 
Set.nonempty_def: ∀ {α : Type ?u.41891} {s : Set α}, Set.Nonempty s ↔ ∃ x, x ∈ s
Set.nonempty_def.
mp: ∀ {a b : Prop}, (a ↔ b) → a → b
mp 
h𝓑: Set.Nonempty 𝓑
h𝓑
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

hx: x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b

hA: x ∈ A

b: Set α

hb: b ∈ 𝓑

h.mp.left
∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)
              
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

hx: x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b

h.mp.left
x ∈ A → ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)exact ⟨
b: Set α
b, ⟨
hb: b ∈ 𝓑
hb, 
Or.inl: ∀ {a b : Prop}, a → a ∨ b
Or.inl 
hA: x ∈ A
hA⟩⟩
Goals accomplished! 🐙
            
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

h.mp
(x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b) → ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)·
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

hx: x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b

h.mp.right
(∃ b, b ∈ 𝓑 ∧ x ∈ b) → ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b) 
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

hx: x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b

h.mp.right
(∃ b, b ∈ 𝓑 ∧ x ∈ b) → ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)intro ⟨
b: Set α
b, 
hb: b ∈ 𝓑 ∧ x ∈ b
hb⟩
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

hx: x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b

b: Set α

hb: b ∈ 𝓑 ∧ x ∈ b

h.mp.right
∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)
              
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

hx: x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b

h.mp.right
(∃ b, b ∈ 𝓑 ∧ x ∈ b) → ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)exact ⟨
b: Set α
b, ⟨
hb: b ∈ 𝓑 ∧ x ∈ b
hb.
left: ∀ {a b : Prop}, a ∧ b → a
left, 
Or.inr: ∀ {a b : Prop}, b → a ∨ b
Or.inr 
hb: b ∈ 𝓑 ∧ x ∈ b
hb.
right: ∀ {a b : Prop}, a ∧ b → b
right⟩⟩
Goals accomplished! 🐙
          
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

{x | x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b} = {x | ∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)}·
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

h.mpr
(∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)) → x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b 
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

h.mpr
(∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)) → x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ bintro ⟨
b: Set α
b, ⟨
hb: b ∈ 𝓑
hb, 
hx: x ∈ A ∨ x ∈ b
hx⟩⟩
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

b: Set α

hb: b ∈ 𝓑

hx: x ∈ A ∨ x ∈ b

h.mpr
x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b
            
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

h.mpr
(∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)) → x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ bapply 
Or.elim: ∀ {a b c : Prop}, a ∨ b → (a → c) → (b → c) → c
Or.elim 
hx: x ∈ A ∨ x ∈ b
hx
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

b: Set α

hb: b ∈ 𝓑

hx: x ∈ A ∨ x ∈ b

h.mpr.left
x ∈ A → x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

b: Set α

hb: b ∈ 𝓑

hx: x ∈ A ∨ x ∈ b

h.mpr.right
x ∈ b → x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b
            
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

h.mpr
(∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)) → x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b·
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

b: Set α

hb: b ∈ 𝓑

hx: x ∈ A ∨ x ∈ b

h.mpr.left
x ∈ A → x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b 
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

b: Set α

hb: b ∈ 𝓑

hx: x ∈ A ∨ x ∈ b

h.mpr.left
x ∈ A → x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

b: Set α

hb: b ∈ 𝓑

hx: x ∈ A ∨ x ∈ b

h.mpr.right
x ∈ b → x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ bexact (
Or.inl: ∀ {a b : Prop}, a → a ∨ b
Or.inl ·)
Goals accomplished! 🐙
            
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

h.mpr
(∃ b, b ∈ 𝓑 ∧ (x ∈ A ∨ x ∈ b)) → x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b·
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

b: Set α

hb: b ∈ 𝓑

hx: x ∈ A ∨ x ∈ b

h.mpr.right
x ∈ b → x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b 
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

b: Set α

hb: b ∈ 𝓑

hx: x ∈ A ∨ x ∈ b

h.mpr.right
x ∈ b → x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ bintro 
h: x ∈ b
h
α: Type u_1

A: Set α

𝓑: Set (Set α)

h✝: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

b: Set α

hb: b ∈ 𝓑

hx: x ∈ A ∨ x ∈ b

h: x ∈ b

h.mpr.right
x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ b
              
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

x: α

b: Set α

hb: b ∈ 𝓑

hx: x ∈ A ∨ x ∈ b

h.mpr.right
x ∈ b → x ∈ A ∨ ∃ b, b ∈ 𝓑 ∧ x ∈ bexact 
Or.inr: ∀ {a b : Prop}, b → a ∨ b
Or.inr ⟨
b: Set α
b, ⟨
hb: b ∈ 𝓑
hb, 
h: x ∈ b
h⟩⟩
Goals accomplished! 🐙
        
_: ?m✝
_ = { 
x: ?m.41412
x | ∃ 
b: ?m.41417
b ∈ 
𝓑: Set (Set α)
𝓑, 
x: ?m.41412
x ∈ 
A: Set α
A ∪ 
b: ?m.41417
b } := 
rfl: ∀ {α : Sort ?u.41463} {a : α}, a = a
rfl
        
_: ?m✝
_ = { 
x: ?m.41535
x | ∃ 
t: ?m.41540
t, 
t: ?m.41540
t ∈ { 
y: ?m.41551
y | ∃ 
X: ?m.41556
X, 
X: ?m.41556
X ∈ 
𝓑: Set (Set α)
𝓑 ∧ 
A: Set α
A ∪ 
X: ?m.41556
X = 
y: ?m.41551
y } ∧ 
x: ?m.41535
x ∈ 
t: ?m.41540
t } := 
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

mpr.right
A ∪ ⋃₀ 𝓑 = ⋃₀ {x | ∃ X, X ∈ 𝓑 ∧ A ∪ X = x}by
Goals accomplished! 🐙 
α: Type u_1

A: Set α

𝓑: Set (Set α)

h: A = ∅ ∨ Set.Nonempty 𝓑

h𝓑: Set.Nonempty 𝓑

{x | ∃ b, b ∈ 𝓑 ∧ x ∈ A ∪ b} = {x | ∃ t, t ∈ {y | ∃ X, X ∈ 𝓑 ∧ A ∪ X = y} ∧ x ∈ t}simp
Goals accomplished! 🐙
        
_: ?m✝
_ = ⋃₀ { 
A: Set α
A ∪ 
X: ?m.41683
X | 
X: ?m.41683
X ∈ 
𝓑: Set (Set α)
𝓑 } := 
rfl: ∀ {α : Sort ?u.41741} {a : α}, a = a
rfl
Goals accomplished! 🐙

end Enderton.Set.Chapter_2